Functions¶
Manipulation and Creation of States and Operators¶
Quantum States¶

basis
(dimensions, n=None, offset=None)[source]¶ Generates the vector representation of a Fock state.
 Parameters
 dimensionsint or list of ints
Number of Fock states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
 nint or list of ints, optional (default 0 for all dimensions)
Integer corresponding to desired number state, defaults to 0 for all dimensions if omitted. The shape must match
dimensions
, e.g. ifdimensions
is a list, thenn
must either be omitted or a list of equal length. offsetint or list of ints, optional (default 0 for all dimensions)
The lowest number state that is included in the finite number state representation of the state in the relevant dimension.
 Returns
 state
qutip.Qobj
Qobj representing the requested number state
n>
.
 state
Notes
A subtle incompatibility with the quantum optics toolbox: In QuTiP:
basis(N, 0) = ground state
but in the qotoolbox:
basis(N, 1) = ground state
Examples
>>> basis(5,2) Quantum object: dims = [[5], [1]], shape = (5, 1), type = ket Qobj data = [[ 0.+0.j] [ 0.+0.j] [ 1.+0.j] [ 0.+0.j] [ 0.+0.j]] >>> basis([2,2,2], [0,1,0]) Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = (8, 1), type = ket Qobj data = [[0.] [0.] [1.] [0.] [0.] [0.] [0.] [0.]]

bra
(seq, dim=2)[source]¶ Produces a multiparticle bra state for a list or string, where each element stands for state of the respective particle.
 Parameters
 seqstr / list of ints or characters
Each element defines state of the respective particle. (e.g. [1,1,0,1] or a string “1101”). For qubits it is also possible to use the following conventions:  ‘g’/’e’ (ground and excited state)  ‘u’/’d’ (spin up and down)  ‘H’/’V’ (horizontal and vertical polarization) Note: for dimension > 9 you need to use a list.
 dimint (default: 2) / list of ints
Space dimension for each particle: int if there are the same, list if they are different.
 Returns
 braqobj
Examples
>>> bra("10") Quantum object: dims = [[1, 1], [2, 2]], shape = [1, 4], type = bra Qobj data = [[ 0. 0. 1. 0.]]
>>> bra("Hue") Quantum object: dims = [[1, 1, 1], [2, 2, 2]], shape = [1, 8], type = bra Qobj data = [[ 0. 1. 0. 0. 0. 0. 0. 0.]]
>>> bra("12", 3) Quantum object: dims = [[1, 1], [3, 3]], shape = [1, 9], type = bra Qobj data = [[ 0. 0. 0. 0. 0. 1. 0. 0. 0.]]
>>> bra("31", [5, 2]) Quantum object: dims = [[1, 1], [5, 2]], shape = [1, 10], type = bra Qobj data = [[ 0. 0. 0. 0. 0. 0. 0. 1. 0. 0.]]

coherent
(N, alpha, offset=0, method='operator')[source]¶ Generates a coherent state with eigenvalue alpha.
Constructed using displacement operator on vacuum state.
 Parameters
 Nint
Number of Fock states in Hilbert space.
 alphafloat/complex
Eigenvalue of coherent state.
 offsetint (default 0)
The lowest number state that is included in the finite number state representation of the state. Using a nonzero offset will make the default method ‘analytic’.
 methodstring {‘operator’, ‘analytic’}
Method for generating coherent state.
 Returns
 stateqobj
Qobj quantum object for coherent state
Notes
Select method ‘operator’ (default) or ‘analytic’. With the ‘operator’ method, the coherent state is generated by displacing the vacuum state using the displacement operator defined in the truncated Hilbert space of size ‘N’. This method guarantees that the resulting state is normalized. With ‘analytic’ method the coherent state is generated using the analytical formula for the coherent state coefficients in the Fock basis. This method does not guarantee that the state is normalized if truncated to a small number of Fock states, but would in that case give more accurate coefficients.
Examples
>>> coherent(5,0.25j) Quantum object: dims = [[5], [1]], shape = [5, 1], type = ket Qobj data = [[ 9.69233235e01+0.j ] [ 0.00000000e+00+0.24230831j] [ 4.28344935e02+0.j ] [ 0.00000000e+000.00618204j] [ 7.80904967e04+0.j ]]

coherent_dm
(N, alpha, offset=0, method='operator')[source]¶ Density matrix representation of a coherent state.
Constructed via outer product of
qutip.states.coherent
 Parameters
 Nint
Number of Fock states in Hilbert space.
 alphafloat/complex
Eigenvalue for coherent state.
 offsetint (default 0)
The lowest number state that is included in the finite number state representation of the state.
 methodstring {‘operator’, ‘analytic’}
Method for generating coherent density matrix.
 Returns
 dmqobj
Density matrix representation of coherent state.
Notes
Select method ‘operator’ (default) or ‘analytic’. With the ‘operator’ method, the coherent density matrix is generated by displacing the vacuum state using the displacement operator defined in the truncated Hilbert space of size ‘N’. This method guarantees that the resulting density matrix is normalized. With ‘analytic’ method the coherent density matrix is generated using the analytical formula for the coherent state coefficients in the Fock basis. This method does not guarantee that the state is normalized if truncated to a small number of Fock states, but would in that case give more accurate coefficients.
Examples
>>> coherent_dm(3,0.25j) Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 0.93941695+0.j 0.000000000.23480733j 0.04216943+0.j ] [ 0.00000000+0.23480733j 0.05869011+0.j 0.000000000.01054025j] [0.04216943+0.j 0.00000000+0.01054025j 0.00189294+0.j ]]

enr_state_dictionaries
(dims, excitations)[source]¶ Return the number of states, and lookupdictionaries for translating a state tuple to a state index, and vice versa, for a system with a given number of components and maximum number of excitations.
 Parameters
 dims: list
A list with the number of states in each subsystem.
 excitationsinteger
The maximum numbers of dimension
 Returns
 nstates, state2idx, idx2state: integer, dict, dict
The number of states nstates, a dictionary for looking up state indices from a state tuple, and a dictionary for looking up state state tuples from state indices.

enr_thermal_dm
(dims, excitations, n)[source]¶ Generate the density operator for a thermal state in the excitationnumber restricted state space defined by the dims and exciations arguments. See the documentation for enr_fock for a more detailed description of these arguments. The temperature of each mode in dims is specified by the average number of excitatons n.
 Parameters
 dimslist
A list of the dimensions of each subsystem of a composite quantum system.
 excitationsinteger
The maximum number of excitations that are to be included in the state space.
 ninteger
The average number of exciations in the thermal state. n can be a float (which then applies to each mode), or a list/array of the same length as dims, in which each element corresponds specifies the temperature of the corresponding mode.
 Returns
 dmQobj
Thermal state density matrix.

enr_fock
(dims, excitations, state)[source]¶ Generate the Fock state representation in a excitationnumber restricted state space. The dims argument is a list of integers that define the number of quantums states of each component of a composite quantum system, and the excitations specifies the maximum number of excitations for the basis states that are to be included in the state space. The state argument is a tuple of integers that specifies the state (in the number basis representation) for which to generate the Fock state representation.
 Parameters
 dimslist
A list of the dimensions of each subsystem of a composite quantum system.
 excitationsinteger
The maximum number of excitations that are to be included in the state space.
 statelist of integers
The state in the number basis representation.
 Returns
 ketQobj
A Qobj instance that represent a Fock state in the exicationnumber restricted state space defined by dims and exciations.

fock
(dimensions, n=None, offset=None)[source]¶ Bosonic Fock (number) state.
Same as
qutip.states.basis
. Parameters
 dimensionsint or list of ints
Number of Fock states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
 nint or list of ints, optional (default 0 for all dimensions)
Integer corresponding to desired number state, defaults to 0 for all dimensions if omitted. The shape must match
dimensions
, e.g. ifdimensions
is a list, thenn
must either be omitted or a list of equal length. offsetint or list of ints, optional (default 0 for all dimensions)
The lowest number state that is included in the finite number state representation of the state in the relevant dimension.
 Returns
 Requested number state \(\leftn\right>\).
Examples
>>> fock(4,3) Quantum object: dims = [[4], [1]], shape = [4, 1], type = ket Qobj data = [[ 0.+0.j] [ 0.+0.j] [ 0.+0.j] [ 1.+0.j]]

fock_dm
(dimensions, n=None, offset=None)[source]¶ Density matrix representation of a Fock state
Constructed via outer product of
qutip.states.fock
. Parameters
 dimensionsint or list of ints
Number of Fock states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
 nint or list of ints, optional (default 0 for all dimensions)
Integer corresponding to desired number state, defaults to 0 for all dimensions if omitted. The shape must match
dimensions
, e.g. ifdimensions
is a list, thenn
must either be omitted or a list of equal length. offsetint or list of ints, optional (default 0 for all dimensions)
The lowest number state that is included in the finite number state representation of the state in the relevant dimension.
 Returns
 dmqobj
Density matrix representation of Fock state.
Examples
>>> fock_dm(3,1) Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 1.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j]]

ghz_state
(N=3)[source]¶ Returns the Nqubit GHZstate.
 Parameters
 Nint (default=3)
Number of qubits in state
 Returns
 Gqobj
Nqubit GHZstate

maximally_mixed_dm
(N)[source]¶ Returns the maximally mixed density matrix for a Hilbert space of dimension N.
 Parameters
 Nint
Number of basis states in Hilbert space.
 Returns
 dmqobj
Thermal state density matrix.

ket
(seq, dim=2)[source]¶ Produces a multiparticle ket state for a list or string, where each element stands for state of the respective particle.
 Parameters
 seqstr / list of ints or characters
Each element defines state of the respective particle. (e.g. [1,1,0,1] or a string “1101”). For qubits it is also possible to use the following conventions:  ‘g’/’e’ (ground and excited state)  ‘u’/’d’ (spin up and down)  ‘H’/’V’ (horizontal and vertical polarization) Note: for dimension > 9 you need to use a list.
 dimint (default: 2) / list of ints
Space dimension for each particle: int if there are the same, list if they are different.
 Returns
 ketqobj
Examples
>>> ket("10") Quantum object: dims = [[2, 2], [1, 1]], shape = [4, 1], type = ket Qobj data = [[ 0.] [ 0.] [ 1.] [ 0.]]
>>> ket("Hue") Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = [8, 1], type = ket Qobj data = [[ 0.] [ 1.] [ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 0.]]
>>> ket("12", 3) Quantum object: dims = [[3, 3], [1, 1]], shape = [9, 1], type = ket Qobj data = [[ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 1.] [ 0.] [ 0.] [ 0.]]
>>> ket("31", [5, 2]) Quantum object: dims = [[5, 2], [1, 1]], shape = [10, 1], type = ket Qobj data = [[ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 1.] [ 0.] [ 0.]]

ket2dm
(Q)[source]¶ Takes input ket or bra vector and returns density matrix formed by outer product.
 Parameters
 Qqobj
Ket or bra type quantum object.
 Returns
 dmqobj
Density matrix formed by outer product of Q.
Examples
>>> x=basis(3,2) >>> ket2dm(x) Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 1.+0.j]]

phase_basis
(N, m, phi0=0)[source]¶ Basis vector for the mth phase of the PeggBarnett phase operator.
 Parameters
 Nint
Number of basis vectors in Hilbert space.
 mint
Integer corresponding to the mth discrete phase phi_m=phi0+2*pi*m/N
 phi0float (default=0)
Reference phase angle.
 Returns
 stateqobj
Ket vector for mth PeggBarnett phase operator basis state.
Notes
The PeggBarnett basis states form a complete set over the truncated Hilbert space.

projection
(N, n, m, offset=None)[source]¶ The projection operator that projects state \(m>\) on state \(n>\).
 Parameters
 Nint
Number of basis states in Hilbert space.
 n, mfloat
The number states in the projection.
 offsetint (default 0)
The lowest number state that is included in the finite number state representation of the projector.
 Returns
 operqobj
Requested projection operator.

qutrit_basis
()[source]¶ Basis states for a three level system (qutrit)
 Returns
 qstatesarray
Array of qutrit basis vectors

singlet_state
()[source]¶ Returns the two particle singletstate:
that is identical to the fourth bell state.
 Returns
 Bell_stateqobj
B11> Bell state

spin_state
(j, m, type='ket')[source]¶ Generates the spin state j, m>, i.e. the eigenstate of the spinj Sz operator with eigenvalue m.
 Parameters
 jfloat
The spin of the state ().
 mint
Eigenvalue of the spinj Sz operator.
 typestring {‘ket’, ‘bra’, ‘dm’}
Type of state to generate.
 Returns
 stateqobj
Qobj quantum object for spin state

spin_coherent
(j, theta, phi, type='ket')[source]¶ Generate the coherent spin state theta, phi>.
 Parameters
 jfloat
The spin of the state.
 thetafloat
Angle from z axis.
 phifloat
Angle from x axis.
 typestring {‘ket’, ‘bra’, ‘dm’}
Type of state to generate.
 Returns
 stateqobj
Qobj quantum object for spin coherent state

state_number_enumerate
(dims, excitations=None, state=None, idx=0)[source]¶ An iterator that enumerate all the state number arrays (quantum numbers on the form [n1, n2, n3, …]) for a system with dimensions given by dims.
Example
>>> for state in state_number_enumerate([2,2]): >>> print(state) [ 0 0 ] [ 0 1 ] [ 1 0 ] [ 1 1 ]
 Parameters
 dimslist or array
The quantum state dimensions array, as it would appear in a Qobj.
 statelist
Current state in the iteration. Used internally.
 excitationsinteger (None)
Restrict state space to states with excitation numbers below or equal to this value.
 idxinteger
Current index in the iteration. Used internally.
 Returns
 state_numberlist
Successive state number arrays that can be used in loops and other iterations, using standard state enumeration by definition.

state_number_index
(dims, state)[source]¶ Return the index of a quantum state corresponding to state, given a system with dimensions given by dims.
Example
>>> state_number_index([2, 2, 2], [1, 1, 0]) 6
 Parameters
 dimslist or array
The quantum state dimensions array, as it would appear in a Qobj.
 statelist
State number array.
 Returns
 idxint
The index of the state given by state in standard enumeration ordering.

state_index_number
(dims, index)[source]¶ Return a quantum number representation given a state index, for a system of composite structure defined by dims.
Example
>>> state_index_number([2, 2, 2], 6) [1, 1, 0]
 Parameters
 dimslist or array
The quantum state dimensions array, as it would appear in a Qobj.
 indexinteger
The index of the state in standard enumeration ordering.
 Returns
 statelist
The state number array corresponding to index index in standard enumeration ordering.

state_number_qobj
(dims, state)[source]¶ Return a Qobj representation of a quantum state specified by the state array state.
Example
>>> state_number_qobj([2, 2, 2], [1, 0, 1]) Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = [8, 1], type = ket Qobj data = [[ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 1.] [ 0.] [ 0.]]
 Parameters
 dimslist or array
The quantum state dimensions array, as it would appear in a Qobj.
 statelist
State number array.
 Returns
 state
qutip.Qobj.qobj
The state as a
qutip.Qobj.qobj
instance.
 state

thermal_dm
(N, n, method='operator')[source]¶ Density matrix for a thermal state of n particles
 Parameters
 Nint
Number of basis states in Hilbert space.
 nfloat
Expectation value for number of particles in thermal state.
 methodstring {‘operator’, ‘analytic’}
string
that sets the method used to generate the thermal state probabilities
 Returns
 dmqobj
Thermal state density matrix.
Notes
The ‘operator’ method (default) generates the thermal state using the truncated number operator
num(N)
. This is the method that should be used in computations. The ‘analytic’ method uses the analytic coefficients derived in an infinite Hilbert space. The analytic form is not necessarily normalized, if truncated too aggressively.Examples
>>> thermal_dm(5, 1) Quantum object: dims = [[5], [5]], shape = [5, 5], type = oper, isHerm = True Qobj data = [[ 0.51612903 0. 0. 0. 0. ] [ 0. 0.25806452 0. 0. 0. ] [ 0. 0. 0.12903226 0. 0. ] [ 0. 0. 0. 0.06451613 0. ] [ 0. 0. 0. 0. 0.03225806]]
>>> thermal_dm(5, 1, 'analytic') Quantum object: dims = [[5], [5]], shape = [5, 5], type = oper, isHerm = True Qobj data = [[ 0.5 0. 0. 0. 0. ] [ 0. 0.25 0. 0. 0. ] [ 0. 0. 0.125 0. 0. ] [ 0. 0. 0. 0.0625 0. ] [ 0. 0. 0. 0. 0.03125]]
Quantum Operators¶
This module contains functions for generating Qobj representation of a variety of commonly occuring quantum operators.

charge
(Nmax, Nmin=None, frac=1)[source]¶ Generate the diagonal charge operator over charge states from Nmin to Nmax.
 Parameters
 Nmaxint
Maximum charge state to consider.
 Nminint (default = Nmax)
Lowest charge state to consider.
 fracfloat (default = 1)
Specify fractional charge if needed.
 Returns
 CQobj
Charge operator over [Nmin,Nmax].
Notes
New in version 3.2.

commutator
(A, B, kind='normal')[source]¶ Return the commutator of kind kind (normal, anti) of the two operators A and B.

create
(N, offset=0)[source]¶ Creation (raising) operator.
 Parameters
 Nint
Dimension of Hilbert space.
 Returns
 operqobj
Qobj for raising operator.
 offsetint (default 0)
The lowest number state that is included in the finite number state representation of the operator.
Examples
>>> create(4) Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isHerm = False Qobj data = [[ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j] [ 1.00000000+0.j 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j] [ 0.00000000+0.j 1.41421356+0.j 0.00000000+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.00000000+0.j 1.73205081+0.j 0.00000000+0.j]]

destroy
(N, offset=0)[source]¶ Destruction (lowering) operator.
 Parameters
 Nint
Dimension of Hilbert space.
 offsetint (default 0)
The lowest number state that is included in the finite number state representation of the operator.
 Returns
 operqobj
Qobj for lowering operator.
Examples
>>> destroy(4) Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isHerm = False Qobj data = [[ 0.00000000+0.j 1.00000000+0.j 0.00000000+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.00000000+0.j 1.41421356+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 1.73205081+0.j] [ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j]]

displace
(N, alpha, offset=0)[source]¶ Singlemode displacement operator.
 Parameters
 Nint
Dimension of Hilbert space.
 alphafloat/complex
Displacement amplitude.
 offsetint (default 0)
The lowest number state that is included in the finite number state representation of the operator.
 Returns
 operqobj
Displacement operator.
Examples
>>> displace(4,0.25) Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isHerm = False Qobj data = [[ 0.96923323+0.j 0.24230859+0.j 0.04282883+0.j 0.00626025+0.j] [ 0.24230859+0.j 0.90866411+0.j 0.33183303+0.j 0.07418172+0.j] [ 0.04282883+0.j 0.33183303+0.j 0.84809499+0.j 0.41083747+0.j] [ 0.00626025+0.j 0.07418172+0.j 0.41083747+0.j 0.90866411+0.j]]

enr_destroy
(dims, excitations)[source]¶ Generate annilation operators for modes in a excitationnumberrestricted state space. For example, consider a system consisting of 4 modes, each with 5 states. The total hilbert space size is 5**4 = 625. If we are only interested in states that contain up to 2 excitations, we only need to include states such as
(0, 0, 0, 0) (0, 0, 0, 1) (0, 0, 0, 2) (0, 0, 1, 0) (0, 0, 1, 1) (0, 0, 2, 0) …
This function creates annihilation operators for the 4 modes that act within this state space:
a1, a2, a3, a4 = enr_destroy([5, 5, 5, 5], excitations=2)
From this point onwards, the annihiltion operators a1, …, a4 can be used to setup a Hamiltonian, collapse operators and expectationvalue operators, etc., following the usual pattern.
 Parameters
 dimslist
A list of the dimensions of each subsystem of a composite quantum system.
 excitationsinteger
The maximum number of excitations that are to be included in the state space.
 Returns
 a_opslist of qobj
A list of annihilation operators for each mode in the composite quantum system described by dims.

enr_identity
(dims, excitations)[source]¶ Generate the identity operator for the excitationnumber restricted state space defined by the dims and exciations arguments. See the docstring for enr_fock for a more detailed description of these arguments.
 Parameters
 dimslist
A list of the dimensions of each subsystem of a composite quantum system.
 excitationsinteger
The maximum number of excitations that are to be included in the state space.
 statelist of integers
The state in the number basis representation.
 Returns
 opQobj
A Qobj instance that represent the identity operator in the exicationnumberrestricted state space defined by dims and exciations.

jmat
(j, *args)[source]¶ Higherorder spin operators:
 Parameters
 jfloat
Spin of operator
 argsstr
Which operator to return ‘x’,’y’,’z’,’+’,’‘. If no args given, then output is [‘x’,’y’,’z’]
 Returns
 jmatqobj / ndarray
qobj
for requested spin operator(s).
Notes
If no ‘args’ input, then returns array of [‘x’,’y’,’z’] operators.
Examples
>>> jmat(1) [ Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 0. 0.70710678 0. ] [ 0.70710678 0. 0.70710678] [ 0. 0.70710678 0. ]] Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 0.+0.j 0.0.70710678j 0.+0.j ] [ 0.+0.70710678j 0.+0.j 0.0.70710678j] [ 0.+0.j 0.+0.70710678j 0.+0.j ]] Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 1. 0. 0.] [ 0. 0. 0.] [ 0. 0. 1.]]]

num
(N, offset=0)[source]¶ Quantum object for number operator.
 Parameters
 Nint
The dimension of the Hilbert space.
 offsetint (default 0)
The lowest number state that is included in the finite number state representation of the operator.
 Returns
 oper: qobj
Qobj for number operator.
Examples
>>> num(4) Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isHerm = True Qobj data = [[0 0 0 0] [0 1 0 0] [0 0 2 0] [0 0 0 3]]

qeye
(dimensions)[source]¶ Identity operator.
 Parameters
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.
 Returns
 operqobj
Identity operator Qobj.
Examples
>>> qeye(3) Quantum object: dims = [[3], [3]], shape = (3, 3), type = oper, isherm = True Qobj data = [[ 1. 0. 0.] [ 0. 1. 0.] [ 0. 0. 1.]] >>> qeye([2,2]) Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True Qobj data = [[1. 0. 0. 0.] [0. 1. 0. 0.] [0. 0. 1. 0.] [0. 0. 0. 1.]]

identity
(dims)[source]¶ Identity operator. Alternative name to
qeye
. Parameters
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.
 Returns
 operqobj
Identity operator Qobj.

momentum
(N, offset=0)[source]¶ Momentum operator p=1j/sqrt(2)*(aa.dag())
 Parameters
 Nint
Number of Fock states in Hilbert space.
 offsetint (default 0)
The lowest number state that is included in the finite number state representation of the operator.
 Returns
 operqobj
Momentum operator as Qobj.

phase
(N, phi0=0)[source]¶ Singlemode PeggBarnett phase operator.
 Parameters
 Nint
Number of basis states in Hilbert space.
 phi0float
Reference phase.
 Returns
 operqobj
Phase operator with respect to reference phase.
Notes
The PeggBarnett phase operator is Hermitian on a truncated Hilbert space.

position
(N, offset=0)[source]¶ Position operator x=1/sqrt(2)*(a+a.dag())
 Parameters
 Nint
Number of Fock states in Hilbert space.
 offsetint (default 0)
The lowest number state that is included in the finite number state representation of the operator.
 Returns
 operqobj
Position operator as Qobj.

qdiags
(diagonals, offsets, dims=None, shape=None)[source]¶ Constructs an operator from an array of diagonals.
 Parameters
 diagonalssequence of array_like
Array of elements to place along the selected diagonals.
 offsetssequence of ints
 Sequence for diagonals to be set:
k=0 main diagonal
k>0 kth upper diagonal
k<0 kth lower diagonal
 dimslist, optional
Dimensions for operator
 shapelist, tuple, optional
Shape of operator. If omitted, a square operator large enough to contain the diagonals is generated.
See also
scipy.sparse.diags
for usage information.
Notes
This function requires SciPy 0.11+.
Examples
>>> qdiags(sqrt(range(1, 4)), 1) Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isherm = False Qobj data = [[ 0. 1. 0. 0. ] [ 0. 0. 1.41421356 0. ] [ 0. 0. 0. 1.73205081] [ 0. 0. 0. 0. ]]

qutrit_ops
()[source]¶ Operators for a three level system (qutrit).
 Returns
 opers: array
array of qutrit operators.

qzero
(dimensions)[source]¶ Zero operator.
 Parameters
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions.
 Returns
 qzeroqobj
Zero operator Qobj.

sigmam
()[source]¶ Annihilation operator for Pauli spins.
Examples
>>> sigmam() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False Qobj data = [[ 0. 0.] [ 1. 0.]]

sigmap
()[source]¶ Creation operator for Pauli spins.
Examples
>>> sigmap() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False Qobj data = [[ 0. 1.] [ 0. 0.]]

sigmax
()[source]¶ Pauli spin 1/2 sigmax operator
Examples
>>> sigmax() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False Qobj data = [[ 0. 1.] [ 1. 0.]]

sigmay
()[source]¶ Pauli spin 1/2 sigmay operator.
Examples
>>> sigmay() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = True Qobj data = [[ 0.+0.j 0.1.j] [ 0.+1.j 0.+0.j]]

sigmaz
()[source]¶ Pauli spin 1/2 sigmaz operator.
Examples
>>> sigmaz() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = True Qobj data = [[ 1. 0.] [ 0. 1.]]

spin_Jx
(j)[source]¶ Spinj x operator
 Parameters
 jfloat
Spin of operator
 Returns
 opQobj
qobj
representation of the operator.

spin_Jy
(j)[source]¶ Spinj y operator
 Parameters
 jfloat
Spin of operator
 Returns
 opQobj
qobj
representation of the operator.

spin_Jz
(j)[source]¶ Spinj z operator
 Parameters
 jfloat
Spin of operator
 Returns
 opQobj
qobj
representation of the operator.

spin_Jm
(j)[source]¶ Spinj annihilation operator
 Parameters
 jfloat
Spin of operator
 Returns
 opQobj
qobj
representation of the operator.

spin_Jp
(j)[source]¶ Spinj creation operator
 Parameters
 jfloat
Spin of operator
 Returns
 opQobj
qobj
representation of the operator.

squeeze
(N, z, offset=0)[source]¶ Singlemode Squeezing operator.
 Parameters
 Nint
Dimension of hilbert space.
 zfloat/complex
Squeezing parameter.
 offsetint (default 0)
The lowest number state that is included in the finite number state representation of the operator.
 Returns
 oper
qutip.qobj.Qobj
Squeezing operator.
 oper
Examples
>>> squeeze(4, 0.25) Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isHerm = False Qobj data = [[ 0.98441565+0.j 0.00000000+0.j 0.17585742+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.95349007+0.j 0.00000000+0.j 0.30142443+0.j] [0.17585742+0.j 0.00000000+0.j 0.98441565+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.30142443+0.j 0.00000000+0.j 0.95349007+0.j]]

squeezing
(a1, a2, z)[source]¶ Generalized squeezing operator.
\[S(z) = \exp\left(\frac{1}{2}\left(z^*a_1a_2  za_1^\dagger a_2^\dagger\right)\right)\] Parameters
 a1
qutip.qobj.Qobj
Operator 1.
 a2
qutip.qobj.Qobj
Operator 2.
 zfloat/complex
Squeezing parameter.
 a1
 Returns
 oper
qutip.qobj.Qobj
Squeezing operator.
 oper
Quantum Objects¶
Random Operators and States¶
This module is a collection of random state and operator generators. The sparsity of the ouput Qobj’s is controlled by varing the density parameter.

rand_dm
(N, density=0.75, pure=False, dims=None, seed=None)[source]¶ Creates a random NxN density matrix.
 Parameters
 Nint, ndarray, list
If int, then shape of output operator. If list/ndarray then eigenvalues of generated density matrix.
 densityfloat
Density between [0,1] of output density matrix.
 dimslist
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[N]].
 Returns
 operqobj
NxN density matrix quantum operator.
Notes
For small density matrices., choosing a low density will result in an error as no diagonal elements will be generated such that \(Tr(\rho)=1\).

rand_dm_ginibre
(N=2, rank=None, dims=None, seed=None)[source]¶ Returns a Ginibre random density operator of dimension
dim
and rankrank
by using the algorithm of [BCSZ08]. Ifrank
is None, a fullrank (HilbertSchmidt ensemble) random density operator will be returned. Parameters
 Nint
Dimension of the density operator to be returned.
 dimslist
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[N]].
 rankint or None
Rank of the sampled density operator. If None, a fullrank density operator is generated.
 Returns
 rhoQobj
An N × N density operator sampled from the Ginibre or HilbertSchmidt distribution.

rand_dm_hs
(N=2, dims=None, seed=None)[source]¶ Returns a HilbertSchmidt random density operator of dimension
dim
and rankrank
by using the algorithm of [BCSZ08]. Parameters
 Nint
Dimension of the density operator to be returned.
 dimslist
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[N]].
 Returns
 rhoQobj
A dim × dim density operator sampled from the Ginibre or HilbertSchmidt distribution.

rand_herm
(N, density=0.75, dims=None, pos_def=False, seed=None)[source]¶ Creates a random NxN sparse Hermitian quantum object.
If ‘N’ is an integer, uses \(H=0.5*(X+X^{+})\) where \(X\) is a randomly generated quantum operator with a given density. Else uses complex Jacobi rotations when ‘N’ is given by an array.
 Parameters
 Nint, list/ndarray
If int, then shape of output operator. If list/ndarray then eigenvalues of generated operator.
 densityfloat
Density between [0,1] of output Hermitian operator.
 dimslist
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[N]].
 pos_defbool (default=False)
Return a positive semidefinite matrix (by diagonal dominance).
 seedint
seed for the random number generator
 Returns
 operqobj
NxN Hermitian quantum operator.

rand_ket
(N=0, density=1, dims=None, seed=None)[source]¶ Creates a random Nx1 sparse ket vector.
 Parameters
 Nint
Number of rows for output quantum operator. If None or 0, N is deduced from dims.
 densityfloat
Density between [0,1] of output ket state.
 dimslist
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[1]].
 Returns
 operqobj
Nx1 ket state quantum operator.

rand_ket_haar
(N=2, dims=None, seed=None)[source]¶ Returns a Haar random pure state of dimension
dim
by applying a Haar random unitary to a fixed pure state. Parameters
 Nint
Dimension of the state vector to be returned. If None or 0, N is deduced from dims.
 dimslist of ints, or None
Dimensions of the resultant quantum object. If None, [[N],[1]] is used.
 Returns
 psiQobj
A random state vector drawn from the Haar measure.

rand_stochastic
(N, density=0.75, kind='left', dims=None, seed=None)[source]¶ Generates a random stochastic matrix.
 Parameters
 Nint
Dimension of matrix.
 densityfloat
Density between [0,1] of output density matrix.
 kindstr (Default = ‘left’)
Generate ‘left’ or ‘right’ stochastic matrix.
 dimslist
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[N]].
 Returns
 operqobj
Quantum operator form of stochastic matrix.

rand_unitary
(N, density=0.75, dims=None, seed=None)[source]¶ Creates a random NxN sparse unitary quantum object.
Uses \(\exp(iH)\) where H is a randomly generated Hermitian operator.
 Parameters
 Nint
Shape of output quantum operator.
 densityfloat
Density between [0,1] of output Unitary operator.
 dimslist
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[N]].
 Returns
 operqobj
NxN Unitary quantum operator.

rand_unitary_haar
(N=2, dims=None, seed=None)[source]¶ Returns a Haar random unitary matrix of dimension
dim
, using the algorithm of [Mez07]. Parameters
 Nint
Dimension of the unitary to be returned.
 dimslist of lists of int, or None
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[N]].
 Returns
 UQobj
Unitary of dims
[[dim], [dim]]
drawn from the Haar measure.

rand_super
(N=5, dims=None, seed=None)[source]¶ Returns a randomly drawn superoperator acting on operators acting on N dimensions.
 Parameters
 Nint
Square root of the dimension of the superoperator to be returned.
 dimslist
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[[N],[N]], [[N],[N]]].

rand_super_bcsz
(N=2, enforce_tp=True, rank=None, dims=None, seed=None)[source]¶ Returns a random superoperator drawn from the Bruzda et al ensemble for CPTP maps [BCSZ08]. Note that due to finite numerical precision, for ranks less than fullrank, zero eigenvalues may become slightly negative, such that the returned operator is not actually completely positive.
 Parameters
 Nint
Square root of the dimension of the superoperator to be returned.
 enforce_tpbool
If True, the tracepreserving condition of [BCSZ08] is enforced; otherwise only complete positivity is enforced.
 rankint or None
Rank of the sampled superoperator. If None, a fullrank superoperator is generated.
 dimslist
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[[N],[N]], [[N],[N]]].
 Returns
 rhoQobj
A superoperator acting on vectorized dim × dim density operators, sampled from the BCSZ distribution.
ThreeLevel Atoms¶
This module provides functions that are useful for simulating the three level atom with QuTiP. A three level atom (qutrit) has three states, which are linked by dipole transitions so that 1 <> 2 <> 3. Depending on there relative energies they are in the ladder, lambda or vee configuration. The structure of the relevant operators is the same for any of the three configurations:
Ladder: Lambda: Vee:
two> three>
three>  
 / \ one> /
 / \  /
 / \ \ /
two> / \ \ /
 / \ \ /
 / \ \ /
 /  \ /
one>  three> 
one> two>
References
The naming of qutip operators follows the convention in [R0be8dcf25d861] .
 R0be8dcf25d861
Shore, B. W., “The Theory of Coherent Atomic Excitation”, Wiley, 1990.
Notes
Contributed by Markus Baden, Oct. 07, 2011
Superoperators and Liouvillians¶

operator_to_vector
(op)[source]¶ Create a vector representation of a quantum operator given the matrix representation.

vector_to_operator
(op)[source]¶ Create a matrix representation given a quantum operator in vector form.

liouvillian
(H, c_ops=[], data_only=False, chi=None)[source]¶ Assembles the Liouvillian superoperator from a Hamiltonian and a
list
of collapse operators. Like liouvillian, but with an experimental implementation which avoids creating extra Qobj instances, which can be advantageous for large systems. Parameters
 HQobj or QobjEvo
System Hamiltonian.
 c_opsarray_like of Qobj or QobjEvo
A
list
orarray
of collapse operators.
 Returns
 LQobj or QobjEvo
Liouvillian superoperator.

spost
(A)[source]¶ Superoperator formed from postmultiplication by operator A
 Parameters
 AQobj or QobjEvo
Quantum operator for post multiplication.
 Returns
 superQobj or QobjEvo
Superoperator formed from input qauntum object.

spre
(A)[source]¶ Superoperator formed from premultiplication by operator A.
 Parameters
 AQobj or QobjEvo
Quantum operator for premultiplication.
 Returns
 super :Qobj or QobjEvo
Superoperator formed from input quantum object.

sprepost
(A, B)[source]¶ Superoperator formed from premultiplication by operator A and post multiplication of operator B.
 Parameters
 AQobj or QobjEvo
Quantum operator for premultiplication.
 BQobj or QobjEvo
Quantum operator for postmultiplication.
 Returns
 superQobj or QobjEvo
Superoperator formed from input quantum objects.

lindblad_dissipator
(a, b=None, data_only=False, chi=None)[source]¶ Lindblad dissipator (generalized) for a single pair of collapse operators (a, b), or for a single collapse operator (a) when b is not specified:
\[\mathcal{D}[a,b]\rho = a \rho b^\dagger  \frac{1}{2}a^\dagger b\rho  \frac{1}{2}\rho a^\dagger b\] Parameters
 aQobj or QobjEvo
Left part of collapse operator.
 bQobj or QobjEvo (optional)
Right part of collapse operator. If not specified, b defaults to a.
 Returns
 Dqobj, QobjEvo
Lindblad dissipator superoperator.
Superoperator Representations¶
This module implements transformations between superoperator representations, including supermatrix, Kraus, Choi and Chi (process) matrix formalisms.

super_to_choi
(q_oper)[source]¶ Takes a superoperator to a Choi matrix TODO: Sanitize input, incorporate as method on Qobj if type==’super’

choi_to_super
(q_oper)[source]¶ Takes a Choi matrix to a superoperator TODO: Sanitize input, Abstractify application of channels to states

choi_to_kraus
(q_oper, tol=1e09)[source]¶ Takes a Choi matrix and returns a list of Kraus operators. TODO: Create a new class structure for quantum channels, perhaps as a strict subclass of Qobj.

kraus_to_choi
(kraus_list)[source]¶ Takes a list of Kraus operators and returns the Choi matrix for the channel represented by the Kraus operators in kraus_list

kraus_to_super
(kraus_list)[source]¶ Converts a list of Kraus operators and returns a super operator.

choi_to_chi
(q_oper)[source]¶ Converts a Choi matrix to a Chi matrix in the Pauli basis.
NOTE: this is only supported for qubits right now. Need to extend to HeisenbergWeyl for other subsystem dimensions.

chi_to_choi
(q_oper)[source]¶ Converts a Choi matrix to a Chi matrix in the Pauli basis.
NOTE: this is only supported for qubits right now. Need to extend to HeisenbergWeyl for other subsystem dimensions.

to_choi
(q_oper)[source]¶ Converts a Qobj representing a quantum map to the Choi representation, such that the trace of the returned operator is equal to the dimension of the system.
 Parameters
 q_operQobj
Superoperator to be converted to Choi representation. If
q_oper
istype="oper"
, then it is taken to act by conjugation, such thatto_choi(A) == to_choi(sprepost(A, A.dag()))
.
 Returns
 choiQobj
A quantum object representing the same map as
q_oper
, such thatchoi.superrep == "choi"
.
 Raises
 TypeError: if the given quantum object is not a map, or cannot be converted
to Choi representation.

to_chi
(q_oper)[source]¶ Converts a Qobj representing a quantum map to a representation as a chi (process) matrix in the Pauli basis, such that the trace of the returned operator is equal to the dimension of the system.
 Parameters
 q_operQobj
Superoperator to be converted to Chi representation. If
q_oper
istype="oper"
, then it is taken to act by conjugation, such thatto_chi(A) == to_chi(sprepost(A, A.dag()))
.
 Returns
 chiQobj
A quantum object representing the same map as
q_oper
, such thatchi.superrep == "chi"
.
 Raises
 TypeError: if the given quantum object is not a map, or cannot be converted
to Chi representation.

to_super
(q_oper)[source]¶ Converts a Qobj representing a quantum map to the supermatrix (Liouville) representation.
 Parameters
 q_operQobj
Superoperator to be converted to supermatrix representation. If
q_oper
istype="oper"
, then it is taken to act by conjugation, such thatto_super(A) == sprepost(A, A.dag())
.
 Returns
 superopQobj
A quantum object representing the same map as
q_oper
, such thatsuperop.superrep == "super"
.
 Raises
 TypeError
If the given quantum object is not a map, or cannot be converted to supermatrix representation.

to_kraus
(q_oper, tol=1e09)[source]¶ Converts a Qobj representing a quantum map to a list of quantum objects, each representing an operator in the Kraus decomposition of the given map.
 Parameters
 q_operQobj
Superoperator to be converted to Kraus representation. If
q_oper
istype="oper"
, then it is taken to act by conjugation, such thatto_kraus(A) == to_kraus(sprepost(A, A.dag())) == [A]
. tolFloat
Optional threshold parameter for eigenvalues/Kraus ops to be discarded. The default is to=1e9.
 Returns
 kraus_opslist of Qobj
A list of quantum objects, each representing a Kraus operator in the decomposition of
q_oper
.
 Raises
 TypeError: if the given quantum object is not a map, or cannot be
decomposed into Kraus operators.

to_stinespring
(q_oper)[source]¶ Converts a Qobj representing a quantum map $Lambda$ to a pair of partial isometries $A$ and $B$ such that $Lambda(X) = Tr_2(A X B^dagger)$ for all inputs $X$, where the partial trace is taken over a a new index on the output dimensions of $A$ and $B$.
For completely positive inputs, $A$ will always equal $B$ up to precision errors.
 Parameters
 q_operQobj
Superoperator to be converted to a Stinespring pair.
 Returns
 A, BQobj
Quantum objects representing each of the Stinespring matrices for the input Qobj.
Operators and Superoperator Dimensions¶
Internal use module for manipulating dims specifications.

is_scalar
(dims)[source]¶ Returns True if a dims specification is effectively a scalar (has dimension 1).

flatten
(l)[source]¶ Flattens a list of lists to the first level.
Given a list containing a mix of scalars and lists, flattens down to a list of the scalars within the original list.
Examples
>>> print(flatten([[[0], 1], 2])) [0, 1, 2]

deep_remove
(l, *what)[source]¶ Removes scalars from all levels of a nested list.
Given a list containing a mix of scalars and lists, returns a list of the same structure, but where one or more scalars have been removed.
Examples
>>> print(deep_remove([[[[0, 1, 2]], [3, 4], [5], [6, 7]]], 0, 5)) [[[[1, 2]], [3, 4], [], [6, 7]]]

unflatten
(l, idxs)[source]¶ Unflattens a list by a given structure.
Given a list of scalars and a deep list of indices as produced by flatten, returns an “unflattened” form of the list. This perfectly inverts flatten.
Examples
>>> l = [[[10, 20, 30], [40, 50, 60]], [[70, 80, 90], [100, 110, 120]]] >>> idxs = enumerate_flat(l) >>> print(unflatten(flatten(l)), idxs) == l True

collapse_dims_oper
(dims)[source]¶ Given the dimensions specifications for a ket, bra or opertype Qobj, returns a dimensions specification describing the same shape by collapsing all composite systems. For instance, the bratype dimensions specification
[[2, 3], [1]]
collapses to[[6], [1]]
. Parameters
 dimslist of lists of ints
Dimensions specifications to be collapsed.
 Returns
 collapsed_dimslist of lists of ints
Collapsed dimensions specification describing the same shape such that
len(collapsed_dims[0]) == len(collapsed_dims[1]) == 1
.

collapse_dims_super
(dims)[source]¶ Given the dimensions specifications for an operatorket, operatorbra or supertype Qobj, returns a dimensions specification describing the same shape by collapsing all composite systems. For instance, the supertype dimensions specification
[[[2, 3], [2, 3]], [[2, 3], [2, 3]]]
collapses to[[[6], [6]], [[6], [6]]]
. Parameters
 dimslist of lists of ints
Dimensions specifications to be collapsed.
 Returns
 collapsed_dimslist of lists of ints
Collapsed dimensions specification describing the same shape such that
len(collapsed_dims[i][j]) == 1
fori
andj
inrange(2)
.

enumerate_flat
(l)[source]¶ Labels the indices at which scalars occur in a flattened list.
Given a list containing a mix of scalars and lists, returns a list of the same structure, where each scalar has been replaced by an index into the flattened list.
Examples
>>> print(enumerate_flat([[[10], [20, 30]], 40])) [[[0], [1, 2]], 3]

dims_to_tensor_perm
(dims)[source]¶ Given the dims of a Qobj instance, returns a list representing a permutation from the flattening of that dims specification to the corresponding tensor indices.
 Parameters
 dimslist
Dimensions specification for a Qobj.
 Returns
 permlist
A list such that
data[flatten(dims)[idx]]
gives the index of the tensordata
corresponding to theidx``th dimension of ``dims
.

dims_to_tensor_shape
(dims)[source]¶ Given the dims of a Qobj instance, returns the shape of the corresponding tensor. This helps, for instance, resolve the columnstacking convention for superoperators.
 Parameters
 dimslist
Dimensions specification for a Qobj.
 Returns
 tensor_shapetuple
NumPy shape of the corresponding tensor.

dims_idxs_to_tensor_idxs
(dims, indices)[source]¶ Given the dims of a Qobj instance, and some indices into dims, returns the corresponding tensor indices. This helps resolve, for instance, that columnstacking for superoperators, operket and operbra implies that the input and output tensor indices are reversed from their order in dims.
 Parameters
 dimslist
Dimensions specification for a Qobj.
 indicesint, list or tuple
Indices to convert to tensor indices. Can be specified as a single index, or as a collection of indices. In the latter case, this can be nested arbitrarily deep. For instance, [0, [0, (2, 3)]].
 Returns
 tens_indicesint, list or tuple
Container of the same structure as indices containing the tensor indices for each element of indices.
Functions acting on states and operators¶
Expectation Values¶

expect
(oper, state)[source]¶ Calculates the expectation value for operator(s) and state(s).
 Parameters
 operqobj/arraylike
A single or a list or operators for expectation value.
 stateqobj/arraylike
A single or a list of quantum states or density matrices.
 Returns
 exptfloat/complex/arraylike
Expectation value.
real
if oper is Hermitian,complex
otherwise. A (nested) array of expectaction values of state or operator are arrays.
Examples
>>> expect(num(4), basis(4, 3)) 3
Tensor¶
Module for the creation of composite quantum objects via the tensor product.

tensor
(*args)[source]¶ Calculates the tensor product of input operators.
 Parameters
 argsarray_like
list
orarray
of quantum objects for tensor product.
 Returns
 objqobj
A composite quantum object.
Examples
>>> tensor([sigmax(), sigmax()]) Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = True Qobj data = [[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j] [ 0.+0.j 0.+0.j 1.+0.j 0.+0.j] [ 0.+0.j 1.+0.j 0.+0.j 0.+0.j] [ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]]

super_tensor
(*args)[source]¶ Calculates the tensor product of input superoperators, by tensoring together the underlying Hilbert spaces on which each vectorized operator acts.
 Parameters
 argsarray_like
list
orarray
of quantum objects withtype="super"
.
 Returns
 objqobj
A composite quantum object.

composite
(*args)[source]¶ Given two or more operators, kets or bras, returns the Qobj corresponding to a composite system over each argument. For ordinary operators and vectors, this is the tensor product, while for superoperators and vectorized operators, this is the columnreshuffled tensor product.
If a mix of Qobjs supported on Hilbert and Liouville spaces are passed in, the former are promoted. Ordinary operators are assumed to be unitaries, and are promoted using
to_super
, while kets and bras are promoted by taking their projectors and usingoperator_to_vector(ket2dm(arg))
.

tensor_contract
(qobj, *pairs)[source]¶ Contracts a qobj along one or more index pairs. Note that this uses dense representations and thus should not be used for very large Qobjs.
 Parameters
 pairstuple
One or more tuples
(i, j)
indicating that thei
andj
dimensions of the original qobj should be contracted.
 Returns
 cqobjQobj
The original Qobj with all named index pairs contracted away.
Partial Transpose¶

partial_transpose
(rho, mask, method='dense')[source]¶ Return the partial transpose of a Qobj instance rho, where mask is an array/list with length that equals the number of components of rho (that is, the length of rho.dims[0]), and the values in mask indicates whether or not the corresponding subsystem is to be transposed. The elements in mask can be boolean or integers 0 or 1, where True/1 indicates that the corresponding subsystem should be tranposed.
 Parameters
 rho
qutip.qobj
A density matrix.
 masklist / array
A mask that selects which subsystems should be transposed.
 methodstr
choice of method, dense or sparse. The default method is dense. The sparse implementation can be faster for large and sparse systems (hundreds of quantum states).
 rho
 Returns
 rho_pr:
qutip.qobj
A density matrix with the selected subsystems transposed.
 rho_pr:
Entropy Functions¶

concurrence
(rho)[source]¶ Calculate the concurrence entanglement measure for a twoqubit state.
 Parameters
 stateqobj
Ket, bra, or density matrix for a twoqubit state.
 Returns
 concurfloat
Concurrence
References

entropy_conditional
(rho, selB, base=2.718281828459045, sparse=False)[source]¶ Calculates the conditional entropy \(S(AB)=S(A,B)S(B)\) of a selected density matrix component.
 Parameters
 rhoqobj
Density matrix of composite object
 selBint/list
Selected components for density matrix B
 base{e,2}
Base of logarithm.
 sparse{False,True}
Use sparse eigensolver.
 Returns
 ent_condfloat
Value of conditional entropy

entropy_linear
(rho)[source]¶ Linear entropy of a density matrix.
 Parameters
 rhoqobj
sensity matrix or ket/bra vector.
 Returns
 entropyfloat
Linear entropy of rho.
Examples
>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1) >>> entropy_linear(rho) 0.5

entropy_mutual
(rho, selA, selB, base=2.718281828459045, sparse=False)[source]¶ Calculates the mutual information S(A:B) between selection components of a system density matrix.
 Parameters
 rhoqobj
Density matrix for composite quantum systems
 selAint/list
int or list of first selected density matrix components.
 selBint/list
int or list of second selected density matrix components.
 base{e,2}
Base of logarithm.
 sparse{False,True}
Use sparse eigensolver.
 Returns
 ent_mutfloat
Mutual information between selected components.

entropy_vn
(rho, base=2.718281828459045, sparse=False)[source]¶ VonNeumann entropy of density matrix
 Parameters
 rhoqobj
Density matrix.
 base{e,2}
Base of logarithm.
 sparse{False,True}
Use sparse eigensolver.
 Returns
 entropyfloat
VonNeumann entropy of rho.
Examples
>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1) >>> entropy_vn(rho,2) 1.0
Density Matrix Metrics¶
This module contains a collection of functions for calculating metrics (distance measures) between states and operators.

fidelity
(A, B)[source]¶ Calculates the fidelity (pseudometric) between two density matrices. See: Nielsen & Chuang, “Quantum Computation and Quantum Information”
 Parameters
 Aqobj
Density matrix or state vector.
 Bqobj
Density matrix or state vector with same dimensions as A.
 Returns
 fidfloat
Fidelity pseudometric between A and B.
Examples
>>> x = fock_dm(5,3) >>> y = coherent_dm(5,1) >>> fidelity(x,y) 0.24104350624628332

tracedist
(A, B, sparse=False, tol=0)[source]¶ Calculates the trace distance between two density matrices.. See: Nielsen & Chuang, “Quantum Computation and Quantum Information”
 Parameters
 Aqobj
Density matrix or state vector.
 Bqobj
Density matrix or state vector with same dimensions as A.
 tolfloat
Tolerance used by sparse eigensolver, if used. (0=Machine precision)
 sparse{False, True}
Use sparse eigensolver.
 Returns
 tracedistfloat
Trace distance between A and B.
Examples
>>> x=fock_dm(5,3) >>> y=coherent_dm(5,1) >>> tracedist(x,y) 0.9705143161472971

bures_dist
(A, B)[source]¶ Returns the Bures distance between two density matrices A & B.
The Bures distance ranges from 0, for states with unit fidelity, to sqrt(2).
 Parameters
 Aqobj
Density matrix or state vector.
 Bqobj
Density matrix or state vector with same dimensions as A.
 Returns
 distfloat
Bures distance between density matrices.

bures_angle
(A, B)[source]¶ Returns the Bures Angle between two density matrices A & B.
The Bures angle ranges from 0, for states with unit fidelity, to pi/2.
 Parameters
 Aqobj
Density matrix or state vector.
 Bqobj
Density matrix or state vector with same dimensions as A.
 Returns
 anglefloat
Bures angle between density matrices.

hilbert_dist
(A, B)[source]¶ Returns the HilbertSchmidt distance between two density matrices A & B.
 Parameters
 Aqobj
Density matrix or state vector.
 Bqobj
Density matrix or state vector with same dimensions as A.
 Returns
 distfloat
HilbertSchmidt distance between density matrices.
Notes
See V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998).

average_gate_fidelity
(oper, target=None)[source]¶ Given a Qobj representing the supermatrix form of a map, returns the average gate fidelity (pseudometric) of that map.
 Parameters
 AQobj
Quantum object representing a superoperator.
 targetQobj
Quantum object representing the target unitary; the inverse is applied before evaluating the fidelity.
 Returns
 fidfloat
Fidelity pseudometric between A and the identity superoperator, or between A and the target superunitary.
Continuous Variables¶
This module contains a collection functions for calculating continuous variable quantities from fockbasis representation of the state of multimode fields.

correlation_matrix
(basis, rho=None)[source]¶ Given a basis set of operators \(\{a\}_n\), calculate the correlation matrix:
\[C_{mn} = \langle a_m a_n \rangle\] Parameters
 basislist
List of operators that defines the basis for the correlation matrix.
 rhoQobj
Density matrix for which to calculate the correlation matrix. If rho is None, then a matrix of correlation matrix operators is returned instead of expectation values of those operators.
 Returns
 corr_matndarray
A 2dimensional array of correlation values or operators.

covariance_matrix
(basis, rho, symmetrized=True)[source]¶ Given a basis set of operators \(\{a\}_n\), calculate the covariance matrix:
\[V_{mn} = \frac{1}{2}\langle a_m a_n + a_n a_m \rangle  \langle a_m \rangle \langle a_n\rangle\]or, if of the optional argument symmetrized=False,
\[V_{mn} = \langle a_m a_n\rangle  \langle a_m \rangle \langle a_n\rangle\] Parameters
 basislist
List of operators that defines the basis for the covariance matrix.
 rhoQobj
Density matrix for which to calculate the covariance matrix.
 symmetrizedbool {True, False}
Flag indicating whether the symmetrized (default) or nonsymmetrized correlation matrix is to be calculated.
 Returns
 corr_matndarray
A 2dimensional array of covariance values.

correlation_matrix_field
(a1, a2, rho=None)[source]¶ Calculates the correlation matrix for given field operators \(a_1\) and \(a_2\). If a density matrix is given the expectation values are calculated, otherwise a matrix with operators is returned.
 Parameters
 a1Qobj
Field operator for mode 1.
 a2Qobj
Field operator for mode 2.
 rhoQobj
Density matrix for which to calculate the covariance matrix.
 Returns
 cov_matndarray
Array of complex numbers or Qobj’s A 2dimensional array of covariance values, or, if rho=0, a matrix of operators.

correlation_matrix_quadrature
(a1, a2, rho=None, g=1.4142135623730951)[source]¶ Calculate the quadrature correlation matrix with given field operators \(a_1\) and \(a_2\). If a density matrix is given the expectation values are calculated, otherwise a matrix with operators is returned.
 Parameters
 a1Qobj
Field operator for mode 1.
 a2Qobj
Field operator for mode 2.
 rhoQobj
Density matrix for which to calculate the covariance matrix.
 gfloat
Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). The value of g is related to the value of hbar in the commutation relation [x, y] = i * hbar via hbar=2/g ** 2 giving the default value hbar=1.
 Returns
 corr_matndarray
Array of complex numbers or Qobj’s A 2dimensional array of covariance values for the field quadratures, or, if rho=0, a matrix of operators.

wigner_covariance_matrix
(a1=None, a2=None, R=None, rho=None, g=1.4142135623730951)[source]¶ Calculates the Wigner covariance matrix \(V_{ij} = \frac{1}{2}(R_{ij} + R_{ji})\), given the quadrature correlation matrix \(R_{ij} = \langle R_{i} R_{j}\rangle  \langle R_{i}\rangle \langle R_{j}\rangle\), where \(R = (q_1, p_1, q_2, p_2)^T\) is the vector with quadrature operators for the two modes.
Alternatively, if R = None, and if annihilation operators a1 and a2 for the two modes are supplied instead, the quadrature correlation matrix is constructed from the annihilation operators before then the covariance matrix is calculated.
 Parameters
 a1Qobj
Field operator for mode 1.
 a2Qobj
Field operator for mode 2.
 Rndarray
The quadrature correlation matrix.
 rhoQobj
Density matrix for which to calculate the covariance matrix.
 gfloat
Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). The value of g is related to the value of hbar in the commutation relation [x, y] = i * hbar via hbar=2/g ** 2 giving the default value hbar=1.
 Returns
 cov_matndarray
A 2dimensional array of covariance values.

logarithmic_negativity
(V, g=1.4142135623730951)[source]¶ Calculates the logarithmic negativity given a symmetrized covariance matrix, see
qutip.continous_variables.covariance_matrix
. Note that the twomode field state that is described by V must be Gaussian for this function to applicable. Parameters
 V2d array
The covariance matrix.
 gfloat
Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). The value of g is related to the value of hbar in the commutation relation [x, y] = i * hbar via hbar=2/g ** 2 giving the default value hbar=1.
 Returns
 Nfloat
The logarithmic negativity for the twomode Gaussian state that is described by the the Wigner covariance matrix V.
Dynamics and TimeEvolution¶
Schrödinger Equation¶
This module provides solvers for the unitary Schrodinger equation.

sesolve
(H, psi0, tlist, e_ops=None, args=None, options=None, progress_bar=None, _safe_mode=True)[source]¶ Schrodinger equation evolution of a state vector or unitary matrix for a given Hamiltonian.
Evolve the state vector (psi0) using a given Hamiltonian (H), by integrating the set of ordinary differential equations that define the system. Alternatively evolve a unitary matrix in solving the Schrodinger operator equation.
The output is either the state vector or unitary matrix at arbitrary points in time (tlist), or the expectation values of the supplied operators (e_ops). If e_ops is a callback function, it is invoked for each time in tlist with time and the state as arguments, and the function does not use any return values. e_ops cannot be used in conjunction with solving the Schrodinger operator equation
 Parameters
 H
qutip.qobj
,qutip.qobjevo
, list, callable system Hamiltonian as a Qobj, list of Qobj and coefficient, QobjEvo, or a callback function for timedependent Hamiltonians. list format and options can be found in QobjEvo’s description.
 psi0
qutip.qobj
initial state vector (ket) or initial unitary operator psi0 = U
 tlistlist / array
list of times for \(t\).
 e_opsNone / list of
qutip.qobj
/ callback function single operator or list of operators for which to evaluate expectation values. For list operator evolution, the overlapse is computed:
tr(e_ops[i].dag()*op(t))
 argsNone / dictionary
dictionary of parameters for timedependent Hamiltonians
 optionsNone /
qutip.Qdeoptions
with options for the ODE solver.
 progress_barNone / BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for showing the progress of the simulation.
 H
 Returns
 output:
qutip.solver
An instance of the class
qutip.solver
, which contains either an array of expectation values for the times specified by tlist, or an array or state vectors corresponding to the times in tlist [if e_ops is an empty list], or nothing if a callback function was given inplace of operators for which to calculate the expectation values.
 output:
Master Equation¶
This module provides solvers for the Lindblad master equation and von Neumann equation.

mesolve
(H, rho0, tlist, c_ops=None, e_ops=None, args=None, options=None, progress_bar=None, _safe_mode=True)[source]¶ Master equation evolution of a density matrix for a given Hamiltonian and set of collapse operators, or a Liouvillian.
Evolve the state vector or density matrix (rho0) using a given Hamiltonian (H) and an [optional] set of collapse operators (c_ops), by integrating the set of ordinary differential equations that define the system. In the absence of collapse operators the system is evolved according to the unitary evolution of the Hamiltonian.
The output is either the state vector at arbitrary points in time (tlist), or the expectation values of the supplied operators (e_ops). If e_ops is a callback function, it is invoked for each time in tlist with time and the state as arguments, and the function does not use any return values.
If either H or the Qobj elements in c_ops are superoperators, they will be treated as direct contributions to the total system Liouvillian. This allows to solve master equations that are not on standard Lindblad form by passing a custom Liouvillian in place of either the H or c_ops elements.
Timedependent operators
For timedependent problems, H and c_ops can be callback functions that takes two arguments, time and args, and returns the Hamiltonian or Liouvillian for the system at that point in time (callback format).
Alternatively, H and c_ops can be a specified in a nestedlist format where each element in the list is a list of length 2, containing an operator (
qutip.qobj
) at the first element and where the second element is either a string (list string format), a callback function (list callback format) that evaluates to the timedependent coefficient for the corresponding operator, or a NumPy array (list array format) which specifies the value of the coefficient to the corresponding operator for each value of t in tlist.Examples
H = [[H0, ‘sin(w*t)’], [H1, ‘sin(2*w*t)’]]
H = [[H0, f0_t], [H1, f1_t]]
where f0_t and f1_t are python functions with signature f_t(t, args).
H = [[H0, np.sin(w*tlist)], [H1, np.sin(2*w*tlist)]]
In the list string format and list callback format, the string expression and the callback function must evaluate to a real or complex number (coefficient for the corresponding operator).
In all cases of timedependent operators, args is a dictionary of parameters that is used when evaluating operators. It is passed to the callback functions as second argument.
Additional options
Additional options to mesolve can be set via the options argument, which should be an instance of
qutip.solver.Options
. Many ODE integration options can be set this way, and the store_states and store_final_state options can be used to store states even though expectation values are requested via the e_ops argument.Note
If an element in the listspecification of the Hamiltonian or the list of collapse operators are in superoperator form it will be added to the total Liouvillian of the problem with out further transformation. This allows for using mesolve for solving master equations that are not on standard Lindblad form.
Note
On using callback function: mesolve transforms all
qutip.qobj
objects to sparse matrices before handing the problem to the integrator function. In order for your callback function to work correctly, pass allqutip.qobj
objects that are used in constructing the Hamiltonian via args. mesolve will check forqutip.qobj
in args and handle the conversion to sparse matrices. All otherqutip.qobj
objects that are not passed via args will be passed on to the integrator in scipy which will raise an NotImplemented exception. Parameters
 H
qutip.Qobj
System Hamiltonian, or a callback function for timedependent Hamiltonians, or alternatively a system Liouvillian.
 rho0
qutip.Qobj
initial density matrix or state vector (ket).
 tlistlist / array
list of times for \(t\).
 c_opsNone / list of
qutip.Qobj
single collapse operator, or list of collapse operators, or a list of Liouvillian superoperators.
 e_opsNone / list of
qutip.Qobj
/ callback function single single operator or list of operators for which to evaluate expectation values.
 argsNone / dictionary
dictionary of parameters for timedependent Hamiltonians and collapse operators.
 optionsNone /
qutip.Options
with options for the solver.
 progress_barNone / BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for showing the progress of the simulation.
 H
 Returns
 result:
qutip.Result
An instance of the class
qutip.Result
, which contains either an array result.expect of expectation values for the times specified by tlist, or an array result.states of state vectors or density matrices corresponding to the times in tlist [if e_ops is an empty list], or nothing if a callback function was given in place of operators for which to calculate the expectation values.
 result:
Monte Carlo Evolution¶

mcsolve
(H, psi0, tlist, c_ops=[], e_ops=[], ntraj=0, args={}, options=None, progress_bar=True, map_func=<function parallel_map at 0xd1fd21a60>, map_kwargs={}, _safe_mode=True)[source]¶ Monte Carlo evolution of a state vector \(\psi \rangle\) for a given Hamiltonian and sets of collapse operators, and possibly, operators for calculating expectation values. Options for the underlying ODE solver are given by the Options class.
mcsolve supports timedependent Hamiltonians and collapse operators using either Python functions of strings to represent timedependent coefficients. Note that, the system Hamiltonian MUST have at least one constant term.
As an example of a timedependent problem, consider a Hamiltonian with two terms
H0
andH1
, whereH1
is timedependent with coefficientsin(w*t)
, and collapse operatorsC0
andC1
, whereC1
is timedependent with coeffcientexp(a*t)
. Here, w and a are constant arguments with valuesW
andA
.Using the Python function timedependent format requires two Python functions, one for each collapse coefficient. Therefore, this problem could be expressed as:
def H1_coeff(t,args): return sin(args['w']*t) def C1_coeff(t,args): return exp(args['a']*t) H = [H0, [H1, H1_coeff]] c_ops = [C0, [C1, C1_coeff]] args={'a': A, 'w': W}
or in String (Cython) format we could write:
H = [H0, [H1, 'sin(w*t)']] c_ops = [C0, [C1, 'exp(a*t)']] args={'a': A, 'w': W}
Constant terms are preferably placed first in the Hamiltonian and collapse operator lists.
 Parameters
 H
qutip.Qobj
,list
System Hamiltonian.
 psi0
qutip.Qobj
Initial state vector
 tlistarray_like
Times at which results are recorded.
 ntrajint
Number of trajectories to run.
 c_ops
qutip.Qobj
,list
single collapse operator or a
list
of collapse operators. e_ops
qutip.Qobj
,list
single operator as Qobj or
list
or equivalent of Qobj operators for calculating expectation values. argsdict
Arguments for timedependent Hamiltonian and collapse operator terms.
 optionsOptions
Instance of ODE solver options.
 progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for showing the progress of the simulation. Set to None to disable the progress bar.
 map_func: function
A map function for managing the calls to the singletrajactory solver.
 map_kwargs: dictionary
Optional keyword arguments to the map_func function.
 H
 Returns
 results
qutip.solver.Result
Object storing all results from the simulation.
Note
It is possible to reuse the random number seeds from a previous run of the mcsolver by passing the output Result object seeds via the Options class, i.e. Options(seeds=prev_result.seeds).
 results
Exponential Series¶

essolve
(H, rho0, tlist, c_op_list, e_ops)[source]¶ Evolution of a state vector or density matrix (rho0) for a given Hamiltonian (H) and set of collapse operators (c_op_list), by expressing the ODE as an exponential series. The output is either the state vector at arbitrary points in time (tlist), or the expectation values of the supplied operators (e_ops).
 Parameters
 Hqobj/function_type
System Hamiltonian.
 rho0
qutip.qobj
Initial state density matrix.
 tlistlist/array
list
of times for \(t\). c_op_listlist of
qutip.qobj
list
ofqutip.qobj
collapse operators. e_opslist of
qutip.qobj
list
ofqutip.qobj
operators for which to evaluate expectation values.
 Returns
 expt_arrayarray
Expectation values of wavefunctions/density matrices for the times specified in
tlist
.
Note
This solver does not support timedependent Hamiltonians. ..

ode2es
(L, rho0)[source]¶ Creates an exponential series that describes the time evolution for the initial density matrix (or state vector) rho0, given the Liouvillian (or Hamiltonian) L.
 Parameters
 Lqobj
Liouvillian of the system.
 rho0qobj
Initial state vector or density matrix.
 Returns
 eseries
qutip.eseries
eseries
represention of the system dynamics.
 eseries
BlochRedfield Master Equation¶

brmesolve
(H, psi0, tlist, a_ops=[], e_ops=[], c_ops=[], args={}, use_secular=True, sec_cutoff=0.1, tol=1e12, spectra_cb=None, options=None, progress_bar=None, _safe_mode=True, verbose=False)[source]¶ Solves for the dynamics of a system using the BlochRedfield master equation, given an input Hamiltonian, Hermitian bathcoupling terms and their associated spectrum functions, as well as possible Lindblad collapse operators.
For timeindependent systems, the Hamiltonian must be given as a Qobj, whereas the bathcoupling terms (a_ops), must be written as a nested list of operator  spectrum function pairs, where the frequency is specified by the w variable.
Example
a_ops = [[a+a.dag(),lambda w: 0.2*(w>=0)]]
For timedependent systems, the Hamiltonian, a_ops, and Lindblad collapse operators (c_ops), can be specified in the QuTiP stringbased timedependent format. For the a_op spectra, the frequency variable must be w, and the string cannot contain any other variables other than the possibility of having a timedependence through the time variable t:
Example
a_ops = [[a+a.dag(), ‘0.2*exp(t)*(w>=0)’]]
It is also possible to use Cubic_Spline objects for timedependence. In the case of a_ops, Cubic_Splines must be passed as a tuple:
Example
a_ops = [ [a+a.dag(), ( f(w), g(t)] ]
where f(w) and g(t) are strings or Cubic_spline objects for the bath spectrum and timedependence, respectively.
Finally, if one has bathcouplimg terms of the form H = f(t)*a + conj[f(t)]*a.dag(), then the correct input format is
Example
a_ops = [ [(a,a.dag()), (f(w), g1(t), g2(t))],… ]
where f(w) is the spectrum of the operators while g1(t) and g2(t) are the timedependence of the operators a and a.dag(), respectively
 Parameters
 HQobj / list
System Hamiltonian given as a Qobj or nested list in stringbased format.
 psi0: Qobj
Initial density matrix or state vector (ket).
 tlistarray_like
List of times for evaluating evolution
 a_opslist
Nested list of Hermitian system operators that couple to the bath degrees of freedom, along with their associated spectra.
 e_opslist
List of operators for which to evaluate expectation values.
 c_opslist
List of system collapse operators, or nested list in stringbased format.
 argsdict
Placeholder for future implementation, kept for API consistency.
 use_secularbool {True}
Use secular approximation when evaluating bathcoupling terms.
 sec_cutofffloat {0.1}
Cutoff for secular approximation.
 tolfloat {qutip.setttings.atol}
Tolerance used for removing small values after basis transformation.
 spectra_cblist
DEPRECIATED. Do not use.
 options
qutip.solver.Options
Options for the solver.
 progress_barBaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for showing the progress of the simulation.
 Returns
 result:
qutip.solver.Result
An instance of the class
qutip.solver.Result
, which contains either an array of expectation values, for operators given in e_ops, or a list of states for the times specified by tlist.
 result:

bloch_redfield_tensor
()¶ Calculates the timeindependent BlochRedfield tensor for a system given a set of operators and corresponding spectral functions that describes the system’s couplingto its environment.
 Parameters
 H
qutip.qobj
System Hamiltonian.
 a_opslist
Nested list of system operators that couple to the environment, and the corresponding bath spectra represented as Python functions.
 spectra_cblist
Depreciated.
 c_opslist
List of system collapse operators.
 use_secularbool {True, False}
Flag that indicates if the secular approximation should be used.
 sec_cutofffloat {0.1}
Threshold for secular approximation.
 atolfloat {qutip.settings.atol}
Threshold for removing small parameters.
 H
 Returns
 R, kets:
qutip.Qobj
, list ofqutip.Qobj
R is the BlochRedfield tensor and kets is a list eigenstates of the Hamiltonian.
 R, kets:

bloch_redfield_solve
(R, ekets, rho0, tlist, e_ops=[], options=None, progress_bar=None)[source]¶ Evolve the ODEs defined by BlochRedfield master equation. The BlochRedfield tensor can be calculated by the function
bloch_redfield_tensor
. Parameters
 R
qutip.qobj
BlochRedfield tensor.
 eketsarray of
qutip.qobj
Array of kets that make up a basis tranformation for the eigenbasis.
 rho0
qutip.qobj
Initial density matrix.
 tlistlist / array
List of times for \(t\).
 e_opslist of
qutip.qobj
/ callback function List of operators for which to evaluate expectation values.
 options
qutip.Qdeoptions
Options for the ODE solver.
 R
 Returns
 output:
qutip.solver
An instance of the class
qutip.solver
, which contains either an array of expectation values for the times specified by tlist.
 output:
Floquet States and FloquetMarkov Master Equation¶

fmmesolve
(H, rho0, tlist, c_ops=[], e_ops=[], spectra_cb=[], T=None, args={}, options=<qutip.solver.Options object at 0x1a2035b7b8>, floquet_basis=True, kmax=5, _safe_mode=True)[source]¶ Solve the dynamics for the system using the FloquetMarkov master equation.
Note
This solver currently does not support multiple collapse operators.
 Parameters
 H
qutip.qobj
system Hamiltonian.
 rho0 / psi0
qutip.qobj
initial density matrix or state vector (ket).
 tlistlist / array
list of times for \(t\).
 c_opslist of
qutip.qobj
list of collapse operators.
 e_opslist of
qutip.qobj
/ callback function list of operators for which to evaluate expectation values.
 spectra_cblist callback functions
List of callback functions that compute the noise power spectrum as a function of frequency for the collapse operators in c_ops.
 Tfloat
The period of the timedependence of the hamiltonian. The default value ‘None’ indicates that the ‘tlist’ spans a single period of the driving.
 argsdictionary
dictionary of parameters for timedependent Hamiltonians and collapse operators.
This dictionary should also contain an entry ‘w_th’, which is the temperature of the environment (if finite) in the energy/frequency units of the Hamiltonian. For example, if the Hamiltonian written in units of 2pi GHz, and the temperature is given in K, use the following conversion
>>> temperature = 25e3 # unit K >>> h = 6.626e34 >>> kB = 1.38e23 >>> args['w_th'] = temperature * (kB / h) * 2 * pi * 1e9
 options
qutip.solver
options for the ODE solver.
 k_maxint
The truncation of the number of sidebands (default 5).
 H
 Returns
 output
qutip.solver
An instance of the class
qutip.solver
, which contains either an array of expectation values for the times specified by tlist.
 output

floquet_modes
(H, T, args=None, sort=False, U=None)[source]¶ Calculate the initial Floquet modes Phi_alpha(0) for a driven system with period T.
Returns a list of
qutip.qobj
instances representing the Floquet modes and a list of corresponding quasienergies, sorted by increasing quasienergy in the interval [pi/T, pi/T]. The optional parameter sort decides if the output is to be sorted in increasing quasienergies or not. Parameters
 H
qutip.qobj
system Hamiltonian, timedependent with period T
 argsdictionary
dictionary with variables required to evaluate H
 Tfloat
The period of the timedependence of the hamiltonian. The default value ‘None’ indicates that the ‘tlist’ spans a single period of the driving.
 U
qutip.qobj
The propagator for the timedependent Hamiltonian with period T. If U is None (default), it will be calculated from the Hamiltonian H using
qutip.propagator.propagator
.
 H
 Returns
 outputlist of kets, list of quasi energies
Two lists: the Floquet modes as kets and the quasi energies.

floquet_modes_t
(f_modes_0, f_energies, t, H, T, args=None)[source]¶ Calculate the Floquet modes at times tlist Phi_alpha(tlist) propagting the initial Floquet modes Phi_alpha(0)
 Parameters
 f_modes_0list of
qutip.qobj
(kets) Floquet modes at \(t\)
 f_energieslist
Floquet energies.
 tfloat
The time at which to evaluate the floquet modes.
 H
qutip.qobj
system Hamiltonian, timedependent with period T
 argsdictionary
dictionary with variables required to evaluate H
 Tfloat
The period of the timedependence of the hamiltonian.
 f_modes_0list of
 Returns
 outputlist of kets
The Floquet modes as kets at time \(t\)

floquet_modes_table
(f_modes_0, f_energies, tlist, H, T, args=None)[source]¶ Precalculate the Floquet modes for a range of times spanning the floquet period. Can later be used as a table to look up the floquet modes for any time.
 Parameters
 f_modes_0list of
qutip.qobj
(kets) Floquet modes at \(t\)
 f_energieslist
Floquet energies.
 tlistarray
The list of times at which to evaluate the floquet modes.
 H
qutip.qobj
system Hamiltonian, timedependent with period T
 Tfloat
The period of the timedependence of the hamiltonian.
 argsdictionary
dictionary with variables required to evaluate H
 f_modes_0list of
 Returns
 outputnested list
A nested list of Floquet modes as kets for each time in tlist

floquet_modes_t_lookup
(f_modes_table_t, t, T)[source]¶ Lookup the floquet mode at time t in the precalculated table of floquet modes in the first period of the timedependence.
 Parameters
 f_modes_table_tnested list of
qutip.qobj
(kets) A lookuptable of Floquet modes at times precalculated by
qutip.floquet.floquet_modes_table
. tfloat
The time for which to evaluate the Floquet modes.
 Tfloat
The period of the timedependence of the hamiltonian.
 f_modes_table_tnested list of
 Returns
 outputnested list
A list of Floquet modes as kets for the time that most closely matching the time t in the supplied table of Floquet modes.

floquet_states
(f_modes_t, f_energies, t)[source]¶ Evaluate the floquet states at time t given the Floquet modes at that time.
 Parameters
 f_modes_tlist of
qutip.qobj
(kets) A list of Floquet modes for time \(t\).
 f_energiesarray
The Floquet energies.
 tfloat
The time for which to evaluate the Floquet states.
 f_modes_tlist of
 Returns
 outputlist
A list of Floquet states for the time \(t\).

floquet_states_t
(f_modes_0, f_energies, t, H, T, args=None)[source]¶ Evaluate the floquet states at time t given the initial Floquet modes.
 Parameters
 f_modes_tlist of
qutip.qobj
(kets) A list of initial Floquet modes (for time \(t=0\)).
 f_energiesarray
The Floquet energies.
 tfloat
The time for which to evaluate the Floquet states.
 H
qutip.qobj
System Hamiltonian, timedependent with period T.
 Tfloat
The period of the timedependence of the hamiltonian.
 argsdictionary
Dictionary with variables required to evaluate H.
 f_modes_tlist of
 Returns
 outputlist
A list of Floquet states for the time \(t\).

floquet_wavefunction
(f_modes_t, f_energies, f_coeff, t)[source]¶ Evaluate the wavefunction for a time t using the Floquet state decompositon, given the Floquet modes at time t.
 Parameters
 f_modes_tlist of
qutip.qobj
(kets) A list of initial Floquet modes (for time \(t=0\)).
 f_energiesarray
The Floquet energies.
 f_coeffarray
The coefficients for Floquet decomposition of the initial wavefunction.
 tfloat
The time for which to evaluate the Floquet states.
 f_modes_tlist of
 Returns
 output
qutip.qobj
The wavefunction for the time \(t\).
 output

floquet_wavefunction_t
(f_modes_0, f_energies, f_coeff, t, H, T, args=None)[source]¶ Evaluate the wavefunction for a time t using the Floquet state decompositon, given the initial Floquet modes.
 Parameters
 f_modes_tlist of
qutip.qobj
(kets) A list of initial Floquet modes (for time \(t=0\)).
 f_energiesarray
The Floquet energies.
 f_coeffarray
The coefficients for Floquet decomposition of the initial wavefunction.
 tfloat
The time for which to evaluate the Floquet states.
 H
qutip.qobj
System Hamiltonian, timedependent with period T.
 Tfloat
The period of the timedependence of the hamiltonian.
 argsdictionary
Dictionary with variables required to evaluate H.
 f_modes_tlist of
 Returns
 output
qutip.qobj
The wavefunction for the time \(t\).
 output

floquet_state_decomposition
(f_states, f_energies, psi)[source]¶ Decompose the wavefunction psi (typically an initial state) in terms of the Floquet states, \(\psi = \sum_\alpha c_\alpha \psi_\alpha(0)\).
 Parameters
 f_stateslist of
qutip.qobj
(kets) A list of Floquet modes.
 f_energiesarray
The Floquet energies.
 psi
qutip.qobj
The wavefunction to decompose in the Floquet state basis.
 f_stateslist of
 Returns
 outputarray
The coefficients \(c_\alpha\) in the Floquet state decomposition.

fsesolve
(H, psi0, tlist, e_ops=[], T=None, args={}, Tsteps=100)[source]¶ Solve the Schrodinger equation using the Floquet formalism.
 Parameters
 H
qutip.qobj.Qobj
System Hamiltonian, timedependent with period T.
 psi0
qutip.qobj
Initial state vector (ket).
 tlistlist / array
list of times for \(t\).
 e_opslist of
qutip.qobj
/ callback function list of operators for which to evaluate expectation values. If this list is empty, the state vectors for each time in tlist will be returned instead of expectation values.
 Tfloat
The period of the timedependence of the hamiltonian.
 argsdictionary
Dictionary with variables required to evaluate H.
 Tstepsinteger
The number of time steps in one driving period for which to precalculate the Floquet modes. Tsteps should be an even number.
 H
 Returns
 output
qutip.solver.Result
An instance of the class
qutip.solver.Result
, which contains either an array of expectation values or an array of state vectors, for the times specified by tlist.
 output

floquet_master_equation_rates
(f_modes_0, f_energies, c_op, H, T, args, J_cb, w_th, kmax=5, f_modes_table_t=None)[source]¶ Calculate the rates and matrix elements for the FloquetMarkov master equation.
 Parameters
 f_modes_0list of
qutip.qobj
(kets) A list of initial Floquet modes.
 f_energiesarray
The Floquet energies.
 c_op
qutip.qobj
The collapse operators describing the dissipation.
 H
qutip.qobj
System Hamiltonian, timedependent with period T.
 Tfloat
The period of the timedependence of the hamiltonian.
 argsdictionary
Dictionary with variables required to evaluate H.
 J_cbcallback functions
A callback function that computes the noise power spectrum, as a function of frequency, associated with the collapse operator c_op.
 w_thfloat
The temperature in units of frequency.
 k_maxint
The truncation of the number of sidebands (default 5).
 f_modes_table_tnested list of
qutip.qobj
(kets) A lookuptable of Floquet modes at times precalculated by
qutip.floquet.floquet_modes_table
(optional).
 f_modes_0list of
 Returns
 outputlist
A list (Delta, X, Gamma, A) containing the matrices Delta, X, Gamma and A used in the construction of the FloquetMarkov master equation.

floquet_master_equation_steadystate
(H, A)[source]¶ Returns the steadystate density matrix (in the floquet basis!) for the FloquetMarkov master equation.
Stochastic Schrödinger Equation and Master Equation¶

ssesolve
(H, psi0, times, sc_ops=[], e_ops=[], _safe_mode=True, args={}, **kwargs)[source]¶ Solve stochastic schrodinger equation. Dispatch to specific solvers depending on the value of the solver keyword argument.
 Parameters
 H
qutip.Qobj
, or time dependent system. System Hamiltonian. Can depend on time, see StochasticSolverOptions help for format.
 psi0
qutip.Qobj
State vector (ket).
 timeslist / array
List of times for \(t\). Must be uniformly spaced.
 sc_opslist of
qutip.Qobj
, or time dependent Qobjs. List of stochastic collapse operators. Each stochastic collapse operator will give a deterministic and stochastic contribution to the eqaution of motion according to how the d1 and d2 functions are defined. Can depend on time, see StochasticSolverOptions help for format.
 e_opslist of
qutip.Qobj
single operator or list of operators for which to evaluate expectation values.
 kwargsdictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.
 H
 Returns
 output:
qutip.solver.Result
An instance of the class
qutip.solver.Result
.
 output:

photocurrent_sesolve
(H, psi0, times, sc_ops=[], e_ops=[], _safe_mode=True, args={}, **kwargs)[source]¶ Solve stochastic schrodinger equation using the photocurrent method.
 Parameters
 H
qutip.Qobj
, or time dependent system. System Hamiltonian. Can depend on time, see StochasticSolverOptions help for format.
 psi0
qutip.Qobj
Initial state vector (ket).
 timeslist / array
List of times for \(t\). Must be uniformly spaced.
 sc_opslist of
qutip.Qobj
, or time dependent Qobjs. List of stochastic collapse operators. Each stochastic collapse operator will give a deterministic and stochastic contribution to the eqaution of motion according to how the d1 and d2 functions are defined. Can depend on time, see StochasticSolverOptions help for format.
 e_opslist of
qutip.Qobj
/ callback function single single operator or list of operators for which to evaluate expectation values.
 kwargsdictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.
 H
 Returns
 output:
qutip.solver.Result
An instance of the class
qutip.solver.Result
.
 output:

smepdpsolve
(H, rho0, times, c_ops, e_ops, **kwargs)[source]¶ A stochastic (piecewse deterministic process) PDP solver for density matrix evolution.
 Parameters
 H
qutip.Qobj
System Hamiltonian.
 rho0
qutip.Qobj
Initial density matrix.
 timeslist / array
List of times for \(t\). Must be uniformly spaced.
 c_opslist of
qutip.Qobj
Deterministic collapse operator which will contribute with a standard Lindblad type of dissipation.
 sc_opslist of
qutip.Qobj
List of stochastic collapse operators. Each stochastic collapse operator will give a deterministic and stochastic contribution to the eqaution of motion according to how the d1 and d2 functions are defined.
 e_opslist of
qutip.Qobj
/ callback function single single operator or list of operators for which to evaluate expectation values.
 kwargsdictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.
 H
 Returns
 output:
qutip.solver.Result
An instance of the class
qutip.solver.Result
.
 output:

smesolve
(H, rho0, times, c_ops=[], sc_ops=[], e_ops=[], _safe_mode=True, args={}, **kwargs)[source]¶ Solve stochastic master equation. Dispatch to specific solvers depending on the value of the solver keyword argument.
 Parameters
 H
qutip.Qobj
, or time dependent system. System Hamiltonian. Can depend on time, see StochasticSolverOptions help for format.
 rho0
qutip.Qobj
Initial density matrix or state vector (ket).
 timeslist / array
List of times for \(t\). Must be uniformly spaced.
 c_opslist of
qutip.Qobj
, or time dependent Qobjs. Deterministic collapse operator which will contribute with a standard Lindblad type of dissipation. Can depend on time, see StochasticSolverOptions help for format.
 sc_opslist of
qutip.Qobj
, or time dependent Qobjs. List of stochastic collapse operators. Each stochastic collapse operator will give a deterministic and stochastic contribution to the eqaution of motion according to how the d1 and d2 functions are defined. Can depend on time, see StochasticSolverOptions help for format.
 e_opslist of
qutip.Qobj
single operator or list of operators for which to evaluate expectation values.
 kwargsdictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.
 H
 Returns
 output:
qutip.solver.Result
An instance of the class
qutip.solver.Result
.
 output:

photocurrent_mesolve
(H, rho0, times, c_ops=[], sc_ops=[], e_ops=[], _safe_mode=True, args={}, **kwargs)[source]¶ Solve stochastic master equation using the photocurrent method.
 Parameters
 H
qutip.Qobj
, or time dependent system. System Hamiltonian. Can depend on time, see StochasticSolverOptions help for format.
 rho0
qutip.Qobj
Initial density matrix or state vector (ket).
 timeslist / array
List of times for \(t\). Must be uniformly spaced.
 c_opslist of
qutip.Qobj
, or time dependent Qobjs. Deterministic collapse operator which will contribute with a standard Lindblad type of dissipation. Can depend on time, see StochasticSolverOptions help for format.
 sc_opslist of
qutip.Qobj
, or time dependent Qobjs. List of stochastic collapse operators. Each stochastic collapse operator will give a deterministic and stochastic contribution to the eqaution of motion according to how the d1 and d2 functions are defined. Can depend on time, see StochasticSolverOptions help for format.
 e_opslist of
qutip.Qobj
/ callback function single single operator or list of operators for which to evaluate expectation values.
 kwargsdictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.
 H
 Returns
 output:
qutip.solver.Result
An instance of the class
qutip.solver.Result
.
 output:

ssepdpsolve
(H, psi0, times, c_ops, e_ops, **kwargs)[source]¶ A stochastic (piecewse deterministic process) PDP solver for wavefunction evolution. For most purposes, use
qutip.mcsolve
instead for quantum trajectory simulations. Parameters
 H
qutip.Qobj
System Hamiltonian.
 psi0
qutip.Qobj
Initial state vector (ket).
 timeslist / array
List of times for \(t\). Must be uniformly spaced.
 c_opslist of
qutip.Qobj
Deterministic collapse operator which will contribute with a standard Lindblad type of dissipation.
 e_opslist of
qutip.Qobj
/ callback function single single operator or list of operators for which to evaluate expectation values.
 kwargsdictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.
 H
 Returns
 output:
qutip.solver.Result
An instance of the class
qutip.solver.Result
.
 output:

stochastic_solvers
()[source]¶  Available solvers for ssesolve and smesolve
 eulermaruyama:
A simple generalization of the Euler method for ordinary differential equations to stochastic differential equations. Only solver which could take noncommuting sc_ops. not tested Order 0.5 Code: ‘eulermaruyama’, ‘euler’, 0.5
 milstein, Order 1.0 strong Taylor scheme:
Better approximate numerical solution to stochastic differential equations. Order strong 1.0 Code: ‘milstein’, 1.0 Numerical Solution of Stochastic Differential Equations Chapter 10.3 Eq. (3.1), By Peter E. Kloeden, Eckhard Platen
 milsteinimp, Order 1.0 implicit strong Taylor scheme:
Implicit milstein scheme for the numerical simulation of stiff stochastic differential equations. Order strong 1.0 Code: ‘milsteinimp’ Numerical Solution of Stochastic Differential Equations Chapter 12.2 Eq. (2.9), By Peter E. Kloeden, Eckhard Platen
 predictorcorrector:
Generalization of the trapezoidal method to stochastic differential equations. More stable than explicit methods. Order strong 0.5, weak 1.0 Only the stochastic part is corrected.
(alpha = 0, eta = 1/2) Code: ‘predcorr’, ‘predictorcorrector’, ‘pceuler’
 Both the deterministic and stochastic part corrected.
(alpha = 1/2, eta = 1/2) Code: ‘pceulerimp’, ‘pceuler2’, ‘predcorr2’
Numerical Solution of Stochastic Differential Equations Chapter 15.5 Eq. (5.4), By Peter E. Kloeden, Eckhard Platen
 platen:
Explicit scheme, create the milstein using finite difference instead of derivatives. Also contain some higher order terms, thus converge better than milstein while staying strong order 1.0. Do not require derivatives, therefore usable for
qutip.stochastic.general_stochastic
Order strong 1.0, weak 2.0 Code: ‘platen’, ‘platen1’, ‘explicit1’ The Theory of Open Quantum Systems Chapter 7 Eq. (7.47), H.P Breuer, F. Petruccione rouchon:
Scheme keeping the positivity of the density matrix. (smesolve only) Order strong 1.0? Code: ‘rouchon’, ‘Rouchon’ Eq. 4 of arXiv:1410.5345 with eta=1 Efficient Quantum Filtering for Quantum Feedback Control Pierre Rouchon, Jason F. Ralph arXiv:1410.5345 [quantph] Phys. Rev. A 91, 012118, (2015)
 taylor1.5, Order 1.5 strong Taylor scheme:
Solver with more terms of the ItoTaylor expansion. Default solver for smesolve and ssesolve. Order strong 1.5 Code: ‘taylor1.5’, ‘taylor15’, 1.5, None Numerical Solution of Stochastic Differential Equations Chapter 10.4 Eq. (4.6), By Peter E. Kloeden, Eckhard Platen
 taylor1.5imp, Order 1.5 implicit strong Taylor scheme:
implicit Taylor 1.5 (alpha = 1/2, beta = doesn’t matter) Order strong 1.5 Code: ‘taylor1.5imp’, ‘taylor15imp’ Numerical Solution of Stochastic Differential Equations Chapter 12.2 Eq. (2.18), By Peter E. Kloeden, Eckhard Platen
 explicit1.5, Explicit Order 1.5 Strong Schemes:
Reproduce the order 1.5 strong Taylor scheme using finite difference instead of derivatives. Slower than taylor15 but usable by
qutip.stochastic.general_stochastic
Order strong 1.5 Code: ‘explicit1.5’, ‘explicit15’, ‘platen15’ Numerical Solution of Stochastic Differential Equations Chapter 11.2 Eq. (2.13), By Peter E. Kloeden, Eckhard Platen taylor2.0, Order 2 strong Taylor scheme:
Solver with more terms of the Stratonovich expansion. Order strong 2.0 Code: ‘taylor2.0’, ‘taylor20’, 2.0 Numerical Solution of Stochastic Differential Equations Chapter 10.5 Eq. (5.2), By Peter E. Kloeden, Eckhard Platen
—All solvers, except taylor2.0, are usable in both smesolve and ssesolve and for both heterodyne and homodyne. taylor2.0 only work for 1 stochastic operator not dependent of time with the homodyne method. The
qutip.stochastic.general_stochastic
only accept derivatives free solvers: [‘euler’, ‘platen’, ‘explicit1.5’]. Available solver for photocurrent_sesolve and photocurrent_mesolve:
Photocurrent use ordinary differential equations between stochastic “jump/collapse”.
 euler:
Euler method for ordinary differential equations between jumps. Only 1 jumps per time interval. Default solver Order 1.0 Code: ‘euler’ Quantum measurement and control Chapter 4, Eq 4.19, 4.40, By Howard M. Wiseman, Gerard J. Milburn
 predictor–corrector:
predictor–corrector method (PECE) for ordinary differential equations. Use poisson distribution to obtain the number of jump at each timestep. Order 2.0 Code: ‘predcorr’

general_stochastic
(state0, times, d1, d2, e_ops=[], m_ops=[], _safe_mode=True, len_d2=1, args={}, **kwargs)[source]¶ Solve stochastic general equation. Dispatch to specific solvers depending on the value of the solver keyword argument.
 Parameters
 state0
qutip.Qobj
Initial state vector (ket) or density matrix as a vector.
 timeslist / array
List of times for \(t\). Must be uniformly spaced.
 d1function, callable class
Function representing the deterministic evolution of the system.
 def d1(time (double), state (as a np.array vector)):
return 1d np.array
 d2function, callable class
Function representing the stochastic evolution of the system.
 def d2(time (double), state (as a np.array vector)):
return 2d np.array (N_sc_ops, len(state0))
 len_d2int
Number of output vector produced by d2
 e_opslist of
qutip.Qobj
single operator or list of operators for which to evaluate expectation values. Must be a superoperator if the state vector is a density matrix.
 kwargsdictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.
 state0
 Returns
 output:
qutip.solver.Result
An instance of the class
qutip.solver.Result
.
 output:
Correlation Functions¶

correlation
(H, state0, tlist, taulist, c_ops, a_op, b_op, solver='me', reverse=False, args={}, options=<qutip.solver.Options object at 0x1a2041a400>)[source]¶ Calculate the twooperator twotime correlation function: \(\left<A(t+\tau)B(t)\right>\) along two time axes using the quantum regression theorem and the evolution solver indicated by the solver parameter.
 Parameters
 HQobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
 state0Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented for the me and es solvers.
 tlistarray_like
list of times for \(t\). tlist must be positive and contain the element 0. When taking steadysteady correlations only one tlist value is necessary, i.e. when \(t \rightarrow \infty\); here tlist is automatically set, ignoring user input.
 taulistarray_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
list of collapse operators, may be timedependent for solver choice of me or mc.
 a_opQobj
operator A.
 b_opQobj
operator B.
 reversebool
If True, calculate \(\left<A(t)B(t+\tau)\right>\) instead of \(\left<A(t+\tau)B(t)\right>\).
 solverstr
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
 optionsOptions
solver options class. ntraj is taken as a twoelement list because the mc correlator calls mcsolve() recursively; by default, ntraj=[20, 100]. mc_corr_eps prevents dividebyzero errors in the mc correlator; by default, mc_corr_eps=1e10.
 Returns
 corr_matarray
An 2dimensional array (matrix) of correlation values for the times specified by tlist (first index) and taulist (second index). If tlist is None, then a 1dimensional array of correlation values is returned instead.
References
See, Gardiner, Quantum Noise, Section 5.2.

correlation_ss
(H, taulist, c_ops, a_op, b_op, solver='me', reverse=False, args={}, options=<qutip.solver.Options object at 0x1a2041a3c8>)[source]¶ Calculate the twooperator twotime correlation function:
\[\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>\]along one time axis (given steadystate initial conditions) using the quantum regression theorem and the evolution solver indicated by the solver parameter.
 Parameters
 HQobj
system Hamiltonian.
 taulistarray_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
list of collapse operators.
 a_opQobj
operator A.
 b_opQobj
operator B.
 reversebool
If True, calculate \(\lim_{t \to \infty} \left<A(t)B(t+\tau)\right>\) instead of \(\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>\).
 solverstr
choice of solver (me for masterequation and es for exponential series).
 optionsOptions
solver options class. ntraj is taken as a twoelement list because the mc correlator calls mcsolve() recursively; by default, ntraj=[20, 100]. mc_corr_eps prevents dividebyzero errors in the mc correlator; by default, mc_corr_eps=1e10.
 Returns
 corr_vecarray
An array of correlation values for the times specified by tlist.
References
See, Gardiner, Quantum Noise, Section 5.2.

correlation_2op_1t
(H, state0, taulist, c_ops, a_op, b_op, solver='me', reverse=False, args={}, options=<qutip.solver.Options object at 0x1a2041a278>)[source]¶ Calculate the twooperator twotime correlation function: \(\left<A(t+\tau)B(t)\right>\) along one time axis using the quantum regression theorem and the evolution solver indicated by the solver parameter.
 Parameters
 HQobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
 state0Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented for the me and es solvers.
 taulistarray_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
list of collapse operators, may be timedependent for solver choice of me or mc.
 a_opQobj
operator A.
 b_opQobj
operator B.
 reversebool {False, True}
If True, calculate \(\left<A(t)B(t+\tau)\right>\) instead of \(\left<A(t+\tau)B(t)\right>\).
 solverstr {‘me’, ‘mc’, ‘es’}
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
 optionsOptions
Solver options class. ntraj is taken as a twoelement list because the mc correlator calls mcsolve() recursively; by default, ntraj=[20, 100]. mc_corr_eps prevents dividebyzero errors in the mc correlator; by default, mc_corr_eps=1e10.
 Returns
 corr_vecndarray
An array of correlation values for the times specified by tlist.
References
See, Gardiner, Quantum Noise, Section 5.2.

correlation_2op_2t
(H, state0, tlist, taulist, c_ops, a_op, b_op, solver='me', reverse=False, args={}, options=<qutip.solver.Options object at 0x1a2041a2b0>)[source]¶ Calculate the twooperator twotime correlation function: \(\left<A(t+\tau)B(t)\right>\) along two time axes using the quantum regression theorem and the evolution solver indicated by the solver parameter.
 Parameters
 HQobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
 state0Qobj
Initial state density matrix \(\rho_0\) or state vector \(\psi_0\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented for the me and es solvers.
 tlistarray_like
list of times for \(t\). tlist must be positive and contain the element 0. When taking steadysteady correlations only one tlist value is necessary, i.e. when \(t \rightarrow \infty\); here tlist is automatically set, ignoring user input.
 taulistarray_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
list of collapse operators, may be timedependent for solver choice of me or mc.
 a_opQobj
operator A.
 b_opQobj
operator B.
 reversebool {False, True}
If True, calculate \(\left<A(t)B(t+\tau)\right>\) instead of \(\left<A(t+\tau)B(t)\right>\).
 solverstr
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
 optionsOptions
solver options class. ntraj is taken as a twoelement list because the mc correlator calls mcsolve() recursively; by default, ntraj=[20, 100]. mc_corr_eps prevents dividebyzero errors in the mc correlator; by default, mc_corr_eps=1e10.
 Returns
 corr_matndarray
An 2dimensional array (matrix) of correlation values for the times specified by tlist (first index) and taulist (second index). If tlist is None, then a 1dimensional array of correlation values is returned instead.
References
See, Gardiner, Quantum Noise, Section 5.2.

correlation_3op_1t
(H, state0, taulist, c_ops, a_op, b_op, c_op, solver='me', args={}, options=<qutip.solver.Options object at 0x1a2041a2e8>)[source]¶ Calculate the threeoperator twotime correlation function: \(\left<A(t)B(t+\tau)C(t)\right>\) along one time axis using the quantum regression theorem and the evolution solver indicated by the solver parameter.
Note: it is not possibly to calculate a physically meaningful correlation of this form where \(\tau<0\).
 Parameters
 HQobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
 rho0Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented for the me and es solvers.
 taulistarray_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
list of collapse operators, may be timedependent for solver choice of me or mc.
 a_opQobj
operator A.
 b_opQobj
operator B.
 c_opQobj
operator C.
 solverstr
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
 optionsOptions
solver options class. ntraj is taken as a twoelement list because the mc correlator calls mcsolve() recursively; by default, ntraj=[20, 100]. mc_corr_eps prevents dividebyzero errors in the mc correlator; by default, mc_corr_eps=1e10.
 Returns
 corr_vecarray
An array of correlation values for the times specified by taulist
References
See, Gardiner, Quantum Noise, Section 5.2.

correlation_3op_2t
(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, solver='me', args={}, options=<qutip.solver.Options object at 0x1a2041a320>)[source]¶ Calculate the threeoperator twotime correlation function: \(\left<A(t)B(t+\tau)C(t)\right>\) along two time axes using the quantum regression theorem and the evolution solver indicated by the solver parameter.
Note: it is not possibly to calculate a physically meaningful correlation of this form where \(\tau<0\).
 Parameters
 HQobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
 rho0Qobj
Initial state density matrix \(\rho_0\) or state vector \(\psi_0\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented for the me and es solvers.
 tlistarray_like
list of times for \(t\). tlist must be positive and contain the element 0. When taking steadysteady correlations only one tlist value is necessary, i.e. when \(t \rightarrow \infty\); here tlist is automatically set, ignoring user input.
 taulistarray_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
list of collapse operators, may be timedependent for solver choice of me or mc.
 a_opQobj
operator A.
 b_opQobj
operator B.
 c_opQobj
operator C.
 solverstr
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
 optionsOptions
solver options class. ntraj is taken as a twoelement list because the mc correlator calls mcsolve() recursively; by default, ntraj=[20, 100]. mc_corr_eps prevents dividebyzero errors in the mc correlator; by default, mc_corr_eps=1e10.
 Returns
 corr_matarray
An 2dimensional array (matrix) of correlation values for the times specified by tlist (first index) and taulist (second index). If tlist is None, then a 1dimensional array of correlation values is returned instead.
References
See, Gardiner, Quantum Noise, Section 5.2.

correlation_4op_1t
(H, state0, taulist, c_ops, a_op, b_op, c_op, d_op, solver='me', args={}, options=<qutip.solver.Options object at 0x1a2041a438>)[source]¶ Calculate the fouroperator twotime correlation function: \(\left<A(t)B(t+\tau)C(t+\tau)D(t)\right>\) along one time axis using the quantum regression theorem and the evolution solver indicated by the solver parameter.
Note: it is not possibly to calculate a physically meaningful correlation of this form where \(\tau<0\).
 Parameters
 HQobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
 rho0Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented for the me and es solvers.
 taulistarray_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
list of collapse operators, may be timedependent for solver choice of me or mc.
 a_opQobj
operator A.
 b_opQobj
operator B.
 c_opQobj
operator C.
 d_opQobj
operator D.
 solverstr
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
 optionsOptions
solver options class. ntraj is taken as a twoelement list because the mc correlator calls mcsolve() recursively; by default, ntraj=[20, 100]. mc_corr_eps prevents dividebyzero errors in the mc correlator; by default, mc_corr_eps=1e10.
 Returns
 corr_vecarray
An array of correlation values for the times specified by taulist.
References
See, Gardiner, Quantum Noise, Section 5.2.
Note
Deprecated in QuTiP 3.1 Use correlation_3op_1t() instead.

correlation_4op_2t
(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, d_op, solver='me', args={}, options=<qutip.solver.Options object at 0x1a2041a470>)[source]¶ Calculate the fouroperator twotime correlation function: \(\left<A(t)B(t+\tau)C(t+\tau)D(t)\right>\) along two time axes using the quantum regression theorem and the evolution solver indicated by the solver parameter.
Note: it is not possibly to calculate a physically meaningful correlation of this form where \(\tau<0\).
 Parameters
 HQobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
 rho0Qobj
Initial state density matrix \(\rho_0\) or state vector \(\psi_0\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented for the me and es solvers.
 tlistarray_like
list of times for \(t\). tlist must be positive and contain the element 0. When taking steadysteady correlations only one tlist value is necessary, i.e. when \(t \rightarrow \infty\); here tlist is automatically set, ignoring user input.
 taulistarray_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
list of collapse operators, may be timedependent for solver choice of me or mc.
 a_opQobj
operator A.
 b_opQobj
operator B.
 c_opQobj
operator C.
 d_opQobj
operator D.
 solverstr
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
 optionsOptions
solver options class. ntraj is taken as a twoelement list because the mc correlator calls mcsolve() recursively; by default, ntraj=[20, 100]. mc_corr_eps prevents dividebyzero errors in the mc correlator; by default, mc_corr_eps=1e10.
 Returns
 corr_matarray
An 2dimensional array (matrix) of correlation values for the times specified by tlist (first index) and taulist (second index). If tlist is None, then a 1dimensional array of correlation values is returned instead.
References
See, Gardiner, Quantum Noise, Section 5.2.

spectrum
(H, wlist, c_ops, a_op, b_op, solver='es', use_pinv=False)[source]¶ Calculate the spectrum of the correlation function \(\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>\), i.e., the Fourier transform of the correlation function:
\[S(\omega) = \int_{\infty}^{\infty} \lim_{t \to \infty} \left<A(t+\tau)B(t)\right> e^{i\omega\tau} d\tau.\]using the solver indicated by the solver parameter. Note: this spectrum is only defined for stationary statistics (uses steady state rho0)
 Parameters
 H
qutip.qobj
system Hamiltonian.
 wlistarray_like
list of frequencies for \(\omega\).
 c_opslist
list of collapse operators.
 a_opQobj
operator A.
 b_opQobj
operator B.
 solverstr
choice of solver (es for exponential series and pi for psuedoinverse).
 use_pinvbool
For use with the pi solver: if True use numpy’s pinv method, otherwise use a generic solver.
 H
 Returns
 spectrumarray
An array with spectrum \(S(\omega)\) for the frequencies specified in wlist.

spectrum_ss
(H, wlist, c_ops, a_op, b_op)[source]¶ Calculate the spectrum of the correlation function \(\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>\), i.e., the Fourier transform of the correlation function:
\[S(\omega) = \int_{\infty}^{\infty} \lim_{t \to \infty} \left<A(t+\tau)B(t)\right> e^{i\omega\tau} d\tau.\]using an eseries based solver Note: this spectrum is only defined for stationary statistics (uses steady state rho0).
 Parameters
 H
qutip.qobj
system Hamiltonian.
 wlistarray_like
list of frequencies for \(\omega\).
 c_opslist of
qutip.qobj
list of collapse operators.
 a_op
qutip.qobj
operator A.
 b_op
qutip.qobj
operator B.
 use_pinvbool
If True use numpy’s pinv method, otherwise use a generic solver.
 H
 Returns
 spectrumarray
An array with spectrum \(S(\omega)\) for the frequencies specified in wlist.

spectrum_pi
(H, wlist, c_ops, a_op, b_op, use_pinv=False)[source]¶ Calculate the spectrum of the correlation function \(\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>\), i.e., the Fourier transform of the correlation function:
\[S(\omega) = \int_{\infty}^{\infty} \lim_{t \to \infty} \left<A(t+\tau)B(t)\right> e^{i\omega\tau} d\tau.\]using a psuedoinverse method. Note: this spectrum is only defined for stationary statistics (uses steady state rho0)
 Parameters
 H
qutip.qobj
system Hamiltonian.
 wlistarray_like
list of frequencies for \(\omega\).
 c_opslist of
qutip.qobj
list of collapse operators.
 a_op
qutip.qobj
operator A.
 b_op
qutip.qobj
operator B.
 use_pinvbool
If True use numpy’s pinv method, otherwise use a generic solver.
 H
 Returns
 spectrumarray
An array with spectrum \(S(\omega)\) for the frequencies specified in wlist.

spectrum_correlation_fft
(tlist, y, inverse=False)[source]¶ Calculate the power spectrum corresponding to a twotime correlation function using FFT.
 Parameters
 tlistarray_like
list/array of times \(t\) which the correlation function is given.
 yarray_like
list/array of correlations corresponding to time delays \(t\).
 inverse: boolean
boolean parameter for using a positive exponent in the Fourier Transform instead. Default is False.
 Returns
 w, Stuple
Returns an array of angular frequencies ‘w’ and the corresponding twosided power spectrum ‘S(w)’.

coherence_function_g1
(H, state0, taulist, c_ops, a_op, solver='me', args={}, options=<qutip.solver.Options object at 0x1a2041a358>)[source]¶ Calculate the normalized firstorder quantum coherence function:
\[g^{(1)}(\tau) = \frac{\langle A^\dagger(\tau)A(0)\rangle} {\sqrt{\langle A^\dagger(\tau)A(\tau)\rangle \langle A^\dagger(0)A(0)\rangle}}\]using the quantum regression theorem and the evolution solver indicated by the solver parameter.
 Parameters
 HQobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
 state0Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented for the me and es solvers.
 taulistarray_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
list of collapse operators, may be timedependent for solver choice of me or mc.
 a_opQobj
operator A.
 solverstr
choice of solver (me for masterequation and es for exponential series).
 optionsOptions
solver options class. ntraj is taken as a twoelement list because the mc correlator calls mcsolve() recursively; by default, ntraj=[20, 100]. mc_corr_eps prevents dividebyzero errors in the mc correlator; by default, mc_corr_eps=1e10.
 Returns
 g1, G1tuple
The normalized and unnormalized secondorder coherence function.

coherence_function_g2
(H, state0, taulist, c_ops, a_op, solver='me', args={}, options=<qutip.solver.Options object at 0x1a2041a390>)[source]¶ Calculate the normalized secondorder quantum coherence function:
\[ g^{(2)}(\tau) = \frac{\langle A^\dagger(0)A^\dagger(\tau)A(\tau)A(0)\rangle} {\langle A^\dagger(\tau)A(\tau)\rangle \langle A^\dagger(0)A(0)\rangle}\]using the quantum regression theorem and the evolution solver indicated by the solver parameter.
 Parameters
 HQobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
 state0Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented for the me and es solvers.
 taulistarray_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
list of collapse operators, may be timedependent for solver choice of me or mc.
 a_opQobj
operator A.
 argsdict
Dictionary of arguments to be passed to solver.
 solverstr
choice of solver (me for masterequation and es for exponential series).
 optionsOptions
solver options class. ntraj is taken as a twoelement list because the mc correlator calls mcsolve() recursively; by default, ntraj=[20, 100]. mc_corr_eps prevents dividebyzero errors in the mc correlator; by default, mc_corr_eps=1e10.
 Returns
 g2, G2tuple
The normalized and unnormalized secondorder coherence function.
Steadystate Solvers¶
Module contains functions for solving for the steady state density matrix of open quantum systems defined by a Liouvillian or Hamiltonian and a list of collapse operators.

steadystate
(A, c_op_list=[], method='direct', solver=None, **kwargs)[source]¶ Calculates the steady state for quantum evolution subject to the supplied Hamiltonian or Liouvillian operator and (if given a Hamiltonian) a list of collapse operators.
If the user passes a Hamiltonian then it, along with the list of collapse operators, will be converted into a Liouvillian operator in Lindblad form.
 Parameters
 Aqobj
A Hamiltonian or Liouvillian operator.
 c_op_listlist
A list of collapse operators.
 solverstr {None, ‘scipy’, ‘mkl’}
Selects the sparse solver to use. Default is autoselect based on the availability of the MKL library.
 methodstr {‘direct’, ‘eigen’, ‘iterativegmres’,
‘iterativelgmres’, ‘iterativebicgstab’, ‘svd’, ‘power’, ‘powergmres’, ‘powerlgmres’, ‘powerbicgstab’}
Method for solving the underlying linear equation. Direct LU solver ‘direct’ (default), sparse eigenvalue problem ‘eigen’, iterative GMRES method ‘iterativegmres’, iterative LGMRES method ‘iterativelgmres’, iterative BICGSTAB method ‘iterativebicgstab’, SVD ‘svd’ (dense), or inversepower method ‘power’. The iterative power methods ‘powergmres’, ‘powerlgmres’, ‘powerbicgstab’ use the same solvers as their direct counterparts.
 return_infobool, optional, default = False
Return a dictionary of solverspecific infomation about the solution and how it was obtained.
 sparsebool, optional, default = True
Solve for the steady state using sparse algorithms. If set to False, the underlying Liouvillian operator will be converted into a dense matrix. Use only for ‘smaller’ systems.
 use_rcmbool, optional, default = False
Use reverse CuthillMckee reordering to minimize fillin in the LU factorization of the Liouvillian.
 use_wbmbool, optional, default = False
Use Weighted Bipartite Matching reordering to make the Liouvillian diagonally dominant. This is useful for iterative preconditioners only, and is set to
True
by default when finding a preconditioner. weightfloat, optional
Sets the size of the elements used for adding the unity trace condition to the linear solvers. This is set to the average abs value of the Liouvillian elements if not specified by the user.
 max_iter_refineint {10}
MKL ONLY. Max. number of iterative refinements to perform.
 scaling_vectorsbool {True, False}
MKL ONLY. Scale matrix to unit norm columns and rows.
 weighted_matchingbool {True, False}
MKL ONLY. Use weighted matching to better condition diagonal.
 x0ndarray, optional
ITERATIVE ONLY. Initial guess for solution vector.
 maxiterint, optional, default=1000
ITERATIVE ONLY. Maximum number of iterations to perform.
 tolfloat, optional, default=1e12
ITERATIVE ONLY. Tolerance used for terminating solver.
 mtolfloat, optional, default=None
ITERATIVE ‘power’ methods ONLY. Tolerance for lu solve method. If None given then max(0.1*tol, 1e15) is used
 matolfloat, optional, default=1e15
ITERATIVE ONLY. Absolute tolerance for lu solve method.
 permc_specstr, optional, default=’COLAMD’
ITERATIVE ONLY. Column ordering used internally by superLU for the ‘direct’ LU decomposition method. Options include ‘COLAMD’ and ‘NATURAL’. If using RCM then this is set to ‘NATURAL’ automatically unless explicitly specified.
 use_precondbool optional, default = False
ITERATIVE ONLY. Use an incomplete sparse LU decomposition as a preconditioner for the ‘iterative’ GMRES and BICG solvers. Speeds up convergence time by orders of magnitude in many cases.
 M{sparse matrix, dense matrix, LinearOperator}, optional
ITERATIVE ONLY. Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning can dramatically improve the rate of convergence for iterative methods. If no preconditioner is given and
use_precond = True
, then one is generated automatically. fill_factorfloat, optional, default = 100
ITERATIVE ONLY. Specifies the fill ratio upper bound (>=1) of the iLU preconditioner. Lower values save memory at the cost of longer execution times and a possible singular factorization.
 drop_tolfloat, optional, default = 1e4
ITERATIVE ONLY. Sets the threshold for the magnitude of preconditioner elements that should be dropped. Can be reduced for a courser factorization at the cost of an increased number of iterations, and a possible singular factorization.
 diag_pivot_threshfloat, optional, default = None
ITERATIVE ONLY. Sets the threshold between [0,1] for which diagonal elements are considered acceptable pivot points when using a preconditioner. A value of zero forces the pivot to be the diagonal element.
 ILU_MILUstr, optional, default = ‘smilu_2’
ITERATIVE ONLY. Selects the incomplete LU decomposition method algoithm used in creating the preconditoner. Should only be used by advanced users.
 Returns
 dmqobj
Steady state density matrix.
 infodict, optional
Dictionary containing solverspecific information about the solution.
Notes
The SVD method works only for dense operators (i.e. small systems).

build_preconditioner
(A, c_op_list=[], **kwargs)[source]¶ Constructs a iLU preconditioner necessary for solving for the steady state density matrix using the iterative linear solvers in the ‘steadystate’ function.
 Parameters
 Aqobj
A Hamiltonian or Liouvillian operator.
 c_op_listlist
A list of collapse operators.
 return_infobool, optional, default = False
Return a dictionary of solverspecific infomation about the solution and how it was obtained.
 use_rcmbool, optional, default = False
Use reverse CuthillMckee reordering to minimize fillin in the LU factorization of the Liouvillian.
 use_wbmbool, optional, default = False
Use Weighted Bipartite Matching reordering to make the Liouvillian diagonally dominant. This is useful for iterative preconditioners only, and is set to
True
by default when finding a preconditioner. weightfloat, optional
Sets the size of the elements used for adding the unity trace condition to the linear solvers. This is set to the average abs value of the Liouvillian elements if not specified by the user.
 methodstr, default = ‘iterative’
Tells the preconditioner what type of Liouvillian to build for iLU factorization. For direct iterative methods use ‘iterative’. For power iterative methods use ‘power’.
 permc_specstr, optional, default=’COLAMD’
Column ordering used internally by superLU for the ‘direct’ LU decomposition method. Options include ‘COLAMD’ and ‘NATURAL’. If using RCM then this is set to ‘NATURAL’ automatically unless explicitly specified.
 fill_factorfloat, optional, default = 100
Specifies the fill ratio upper bound (>=1) of the iLU preconditioner. Lower values save memory at the cost of longer execution times and a possible singular factorization.
 drop_tolfloat, optional, default = 1e4
Sets the threshold for the magnitude of preconditioner elements that should be dropped. Can be reduced for a courser factorization at the cost of an increased number of iterations, and a possible singular factorization.
 diag_pivot_threshfloat, optional, default = None
Sets the threshold between [0,1] for which diagonal elements are considered acceptable pivot points when using a preconditioner. A value of zero forces the pivot to be the diagonal element.
 ILU_MILUstr, optional, default = ‘smilu_2’
Selects the incomplete LU decomposition method algoithm used in creating the preconditoner. Should only be used by advanced users.
 Returns
 luobject
Returns a SuperLU object representing iLU preconditioner.
 infodict, optional
Dictionary containing solverspecific information.
Propagators¶

propagator
(H, t, c_op_list=[], args={}, options=None, unitary_mode='batch', parallel=False, progress_bar=None, _safe_mode=True, **kwargs)[source]¶ Calculate the propagator U(t) for the density matrix or wave function such that \(\psi(t) = U(t)\psi(0)\) or \(\rho_{\mathrm vec}(t) = U(t) \rho_{\mathrm vec}(0)\) where \(\rho_{\mathrm vec}\) is the vector representation of the density matrix.
 Parameters
 Hqobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and coefficients in the liststring or listfunction format for timedependent Hamiltonians (see description in
qutip.mesolve
). tfloat or arraylike
Time or list of times for which to evaluate the propagator.
 c_op_listlist
List of qobj collapse operators.
 argslist/array/dictionary
Parameters to callback functions for timedependent Hamiltonians and collapse operators.
 options
qutip.Options
with options for the ODE solver.
 unitary_mode = str (‘batch’, ‘single’)
Solve all basis vectors simulaneously (‘batch’) or individually (‘single’).
 parallelbool {False, True}
Run the propagator in parallel mode. This will override the unitary_mode settings if set to True.
 progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for showing the progress of the simulation. By default no progress bar is used, and if set to True a TextProgressBar will be used.
 Returns
 aqobj
Instance representing the propagator \(U(t)\).
Timedependent problems¶

rhs_generate
(H, c_ops, args={}, options=<qutip.solver.Options object at 0xd1fd3bdd8>, name=None, cleanup=True)[source]¶ Generates the Cython functions needed for solving the dynamics of a given system using the mesolve function inside a parfor loop.
 Parameters
 Hqobj
System Hamiltonian.
 c_opslist
list
of collapse operators. argsdict
Arguments for timedependent Hamiltonian and collapse operator terms.
 optionsOptions
Instance of ODE solver options.
 name: str
Name of generated RHS
 cleanup: bool
Whether the generated cython file should be automatically removed or not.
Notes
Using this function with any solver other than the mesolve function will result in an error.
Scattering in Quantum Optical Systems¶
Photon scattering in quantum optical systems
This module includes a collection of functions for numerically computing photon scattering in driven arbitrary systems coupled to some configuration of output waveguides. The implementation of these functions closely follows the mathematical treatment given in K.A. Fischer, et. al., Scattering of Coherent Pulses from Quantum Optical Systems (2017, arXiv:1710.02875).

temporal_basis_vector
(waveguide_emission_indices, n_time_bins)[source]¶ Generate a temporal basis vector for emissions at specified time bins into specified waveguides.
 Parameters
 waveguide_emission_indiceslist or tuple
List of indices where photon emission occurs for each waveguide, e.g. [[t1_wg1], [t1_wg2, t2_wg2], [], [t1_wg4, t2_wg4, t3_wg4]].
 n_time_binsint
Number of time bins; the range over which each index can vary.
 Returns
 temporal_basis_vector:class: qutip.Qobj
A basis vector representing photon scattering at the specified indices. If there are W waveguides, T times, and N photon emissions, then the basis vector has dimensionality (W*T)^N.

temporal_scattered_state
(H, psi0, n_emissions, c_ops, tlist, system_zero_state=None, construct_effective_hamiltonian=True)[source]¶ Compute the scattered nphoton state projected onto the temporal basis.
 Parameters
 H:class: qutip.Qobj or list
Systemwaveguide(s) Hamiltonian or effective Hamiltonian in Qobj or listcallback format. If construct_effective_hamiltonian is not specified, an effective Hamiltonian is constructed from H and c_ops.
 psi0:class: qutip.Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\).
 n_emissionsint
Number of photon emissions to calculate.
 c_opslist
List of collapse operators for each waveguide; these are assumed to include spontaneous decay rates, e.g. \(\sigma = \sqrt \gamma \cdot a\)
 tlistarray_like
List of times for \(\tau_i\). tlist should contain 0 and exceed the pulse duration / temporal region of interest.
 system_zero_state:class: qutip.Qobj
State representing zero excitations in the system. Defaults to \(\psi(t_0)\)
 construct_effective_hamiltonianbool
Whether an effective Hamiltonian should be constructed from H and c_ops: \(H_{eff} = H  \frac{i}{2} \sum_n \sigma_n^\dagger \sigma_n\) Default: True.
 Returns
 phi_n:class: qutip.Qobj
The scattered bath state projected onto the temporal basis given by tlist. If there are W waveguides, T times, and N photon emissions, then the state is a tensor product state with dimensionality T^(W*N).

scattering_probability
(H, psi0, n_emissions, c_ops, tlist, system_zero_state=None, construct_effective_hamiltonian=True)[source]¶ Compute the integrated probability of scattering n photons in an arbitrary system. This function accepts a nonlinearly spaced array of times.
 Parameters
 H:class: qutip.Qobj or list
Systemwaveguide(s) Hamiltonian or effective Hamiltonian in Qobj or listcallback format. If construct_effective_hamiltonian is not specified, an effective Hamiltonian is constructed from H and c_ops.
 psi0:class: qutip.Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\).
 n_emissionsint
Number of photons emitted by the system (into any combination of waveguides).
 c_opslist
List of collapse operators for each waveguide; these are assumed to include spontaneous decay rates, e.g. \(\sigma = \sqrt \gamma \cdot a\).
 tlistarray_like
List of times for \(\tau_i\). tlist should contain 0 and exceed the pulse duration / temporal region of interest; tlist need not be linearly spaced.
 system_zero_state:class: qutip.Qobj
State representing zero excitations in the system. Defaults to basis(systemDims, 0).
 construct_effective_hamiltonianbool
Whether an effective Hamiltonian should be constructed from H and c_ops: \(H_{eff} = H  \frac{i}{2} \sum_n \sigma_n^\dagger \sigma_n\) Default: True.
 Returns
 scattering_probfloat
The probability of scattering n photons from the system over the time range specified.
Permutational Invariance¶
Permutational Invariant Quantum Solver (PIQS)
This module calculates the Liouvillian for the dynamics of ensembles of identical twolevel systems (TLS) in the presence of local and collective processes by exploiting permutational symmetry and using the Dicke basis. It also allows to characterize nonlinear functions of the density matrix.

num_dicke_ladders
(N)[source]¶ Calculate the total number of ladders in the Dicke space.
For a collection of N twolevel systems it counts how many different “j” exist or the number of blocks in the blockdiagonal matrix.
 Parameters
 N: int
The number of twolevel systems.
 Returns
 Nj: int
The number of Dicke ladders.

num_tls
(nds)[source]¶ Calculate the number of twolevel systems.
 Parameters
 nds: int
The number of Dicke states.
 Returns
 N: int
The number of twolevel systems.

isdiagonal
(mat)[source]¶ Check if the input matrix is diagonal.
 Parameters
 mat: ndarray/Qobj
A 2D numpy array
 Returns
 diag: bool
True/False depending on whether the input matrix is diagonal.

dicke_blocks
(rho)[source]¶ Create the list of blocks for blockdiagonal density matrix in the Dicke basis.
 Parameters
 rho
qutip.Qobj
A 2D blockdiagonal matrix of ones with dimension (nds,nds), where nds is the number of Dicke states for N twolevel systems.
 rho
 Returns
 square_blocks: list of np.array
Give back the blocks list.

dicke_blocks_full
(rho)[source]¶ Give the full (2^Ndimensional) list of blocks for a Dickebasis matrix.
 Parameters
 rho
qutip.Qobj
A 2D blockdiagonal matrix of ones with dimension (nds,nds), where nds is the number of Dicke states for N twolevel systems.
 rho
 Returns
 full_blockslist
The list of blocks expanded in the 2^N space for N qubits.

dicke_function_trace
(f, rho)[source]¶ Calculate the trace of a function on a Dicke density matrix. :param f: A Taylorexpandable function of rho. :type f: function :param rho: A density matrix in the Dicke basis. :type rho:
qutip.Qobj
 Returns
 resfloat
Trace of a nonlinear function on rho.

purity_dicke
(rho)[source]¶ Calculate purity of a density matrix in the Dicke basis. It accounts for the degenerate blocks in the density matrix.
 Parameters
 rho
qutip.Qobj
Density matrix in the Dicke basis of qutip.piqs.jspin(N), for N spins.
 rho
 Returns
 purityfloat
The purity of the quantum state. It’s 1 for pure states, 0<=purity<1 for mixed states.

entropy_vn_dicke
(rho)[source]¶ Von Neumann Entropy of a Dickebasis density matrix.
 Parameters
 rho
qutip.Qobj
A 2D blockdiagonal matrix of ones with dimension (nds,nds), where nds is the number of Dicke states for N twolevel systems.
 rho
 Returns
 entropy_dm: float
Entropy. Use degeneracy to multiply each block.

state_degeneracy
(N, j)[source]¶ Calculate the degeneracy of the Dicke state.
Each state \(j, m\rangle\) includes D(N,j) irreducible representations \(j, m, \alpha\rangle\).
Uses Decimals to calculate higher numerator and denominators numbers.
 Parameters
 N: int
The number of twolevel systems.
 j: float
Total spin eigenvalue (cooperativity).
 Returns
 degeneracy: int
The state degeneracy.

m_degeneracy
(N, m)[source]¶ Calculate the number of Dicke states \(j, m\rangle\) with same energy.
 Parameters
 N: int
The number of twolevel systems.
 m: float
Total spin zaxis projection eigenvalue (proportional to the total energy).
 Returns
 degeneracy: int
The mdegeneracy.

energy_degeneracy
(N, m)[source]¶ Calculate the number of Dicke states with same energy.
The use of the Decimals class allows to explore N > 1000, unlike the builtin function scipy.special.binom
 Parameters
 N: int
The number of twolevel systems.
 m: float
Total spin zaxis projection eigenvalue. This is proportional to the total energy.
 Returns
 degeneracy: int
The energy degeneracy

ap
(j, m)[source]¶ Calculate the coefficient ap by applying J_+ j, m>.
The action of ap is given by: \(J_{+}j, m\rangle = A_{+}(j, m)j, m+1\rangle\)
 Parameters
 j, m: float
The value for j and m in the dicke basis j,m>.
 Returns
 a_plus: float
The value of \(a_{+}\).

am
(j, m)[source]¶ Calculate the operator am used later.
The action of ap is given by: J_{}j, m> = A_{}(jm)j, m1>
 Parameters
 j: float
The value for j.
 m: float
The value for m.
 Returns
 a_minus: float
The value of \(a_{}\).

spin_algebra
(N, op=None)[source]¶ Create the list [sx, sy, sz] with the spin operators.
The operators are constructed for a collection of N twolevel systems (TLSs). Each element of the list, i.e., sx, is a vector of qutip.Qobj objects (spin matrices), as it cointains the list of the SU(2) Pauli matrices for the N TLSs. Each TLS operator sx[i], with i = 0, …, (N1), is placed in a \(2^N\)dimensional Hilbert space.
 Parameters
 N: int
The number of twolevel systems.
 Returns
 spin_operators: list or :class: qutip.Qobj
A list of qutip.Qobj operators  [sx, sy, sz] or the requested operator.
Notes
sx[i] is \(\frac{\sigma_x}{2}\) in the composite Hilbert space.

jspin
(N, op=None, basis='dicke')[source]¶ Calculate the list of collective operators of the total algebra.
The Dicke basis \(j,m\rangle\langle j,m'\) is used by default. Otherwise with “uncoupled” the operators are in a \(2^N\) space.
 Parameters
 N: int
Number of twolevel systems.
 op: str
The operator to return ‘x’,’y’,’z’,’+’,’‘. If no operator given, then output is the list of operators for [‘x’,’y’,’z’].
 basis: str
The basis of the operators  “dicke” or “uncoupled” default: “dicke”.
 Returns
 j_alg: list or :class: qutip.Qobj
A list of qutip.Qobj representing all the operators in the “dicke” or “uncoupled” basis or a single operator requested.

collapse_uncoupled
(N, emission=0.0, dephasing=0.0, pumping=0.0, collective_emission=0.0, collective_dephasing=0.0, collective_pumping=0.0)[source]¶ Create the collapse operators (c_ops) of the Lindbladian in the uncoupled basis
These operators are in the uncoupled basis of the twolevel system (TLS) SU(2) Pauli matrices.
 Parameters
 N: int
The number of twolevel systems.
 emission: float
Incoherent emission coefficient (also nonradiative emission). default: 0.0
 dephasing: float
Local dephasing coefficient. default: 0.0
 pumping: float
Incoherent pumping coefficient. default: 0.0
 collective_emission: float
Collective (superradiant) emmission coefficient. default: 0.0
 collective_pumping: float
Collective pumping coefficient. default: 0.0
 collective_dephasing: float
Collective dephasing coefficient. default: 0.0
 Returns
 c_ops: list
The list of collapse operators as qutip.Qobj for the system.
Notes
The collapse operator list can be given to qutip.mesolve. Notice that the operators are placed in a Hilbert space of dimension \(2^N\). Thus the method is suitable only for small N (of the order of 10).

dicke_basis
(N, jmm1=None)[source]¶ Initialize the density matrix of a Dicke state for several (j, m, m1).
This function can be used to build arbitrary states in the Dicke basis \(j, m\rangle \langle j, m^{\prime}\). We create coefficients for each (j, m, m1) value in the dictionary jmm1. The mapping for the (i, k) index of the density matrix to the j, m> values is given by the cythonized function jmm1_dictionary. A density matrix is created from the given dictionary of coefficients for each (j, m, m1).
 Parameters
 N: int
The number of twolevel systems.
 jmm1: dict
A dictionary of {(j, m, m1): p} that gives a density p for the (j, m, m1) matrix element.
 Returns
 rho: :class: qutip.Qobj
The density matrix in the Dicke basis.

dicke
(N, j, m)[source]¶ Generate a Dicke state as a pure density matrix in the Dicke basis.
For instance, the superradiant state given by \(j, m\rangle = 1, 0\rangle\) for N = 2, and the state is represented as a density matrix of size (nds, nds) or (4, 4), with the (1, 1) element set to 1.
 Parameters
 N: int
The number of twolevel systems.
 j: float
The eigenvalue j of the Dicke state (j, m).
 m: float
The eigenvalue m of the Dicke state (j, m).
 Returns
 rho: :class: qutip.Qobj
The density matrix.

excited
(N, basis='dicke')[source]¶ Generate the density matrix for the excited state.
This state is given by (N/2, N/2) in the default Dicke basis. If the argument basis is “uncoupled” then it generates the state in a 2**N dim Hilbert space.
 Parameters
 N: int
The number of twolevel systems.
 basis: str
The basis to use. Either “dicke” or “uncoupled”.
 Returns
 state: :class: qutip.Qobj
The excited state density matrix in the requested basis.

superradiant
(N, basis='dicke')[source]¶ Generate the density matrix of the superradiant state.
This state is given by (N/2, 0) or (N/2, 0.5) in the Dicke basis. If the argument basis is “uncoupled” then it generates the state in a 2**N dim Hilbert space.
 Parameters
 N: int
The number of twolevel systems.
 basis: str
The basis to use. Either “dicke” or “uncoupled”.
 Returns
 state: :class: qutip.Qobj
The superradiant state density matrix in the requested basis.

css
(N, x=0.7071067811865475, y=0.7071067811865475, basis='dicke', coordinates='cartesian')[source]¶ Generate the density matrix of the Coherent Spin State (CSS).
It can be defined as, \(CSS \rangle = \prod_i^N(a1\rangle_i + b0\rangle_i)\) with \(a = sin(\frac{\theta}{2})\), \(b = e^{i \phi}\cos(\frac{\theta}{2})\). The default basis is that of Dicke space \(j, m\rangle \langle j, m'\). The default state is the symmetric CSS, \(CSS\rangle = +\rangle\).
 Parameters
 N: int
The number of twolevel systems.
 x, y: float
The coefficients of the CSS state.
 basis: str
The basis to use. Either “dicke” or “uncoupled”.
 coordinates: str
Either “cartesian” or “polar”. If polar then the coefficients are constructed as sin(x/2), cos(x/2)e^(iy).
 Returns
 rho: :class: qutip.Qobj
The CSS state density matrix.

ghz
(N, basis='dicke')[source]¶ Generate the density matrix of the GHZ state.
If the argument basis is “uncoupled” then it generates the state in a \(2^N\)dimensional Hilbert space.
 Parameters
 N: int
The number of twolevel systems.
 basis: str
The basis to use. Either “dicke” or “uncoupled”.
 Returns
 state: :class: qutip.Qobj
The GHZ state density matrix in the requested basis.

ground
(N, basis='dicke')[source]¶ Generate the density matrix of the ground state.
This state is given by (N/2, N/2) in the Dicke basis. If the argument basis is “uncoupled” then it generates the state in a \(2^N\)dimensional Hilbert space.
 Parameters
 N: int
The number of twolevel systems.
 basis: str
The basis to use. Either “dicke” or “uncoupled”
 Returns
 state: :class: qutip.Qobj
The ground state density matrix in the requested basis.

identity_uncoupled
(N)[source]¶ Generate the identity in a \(2^N\)dimensional Hilbert space.
The identity matrix is formed from the tensor product of N TLSs.
 Parameters
 N: int
The number of twolevel systems.
 Returns
 identity: :class: qutip.Qobj
The identity matrix.

block_matrix
(N, elements='ones')[source]¶ Construct the blockdiagonal matrix for the Dicke basis.
 Parameters
 Nint
Number of twolevel systems.
 elementsstr {‘ones’ (default),’degeneracy’}
 Returns
 block_matrndarray
A 2D blockdiagonal matrix with dimension (nds,nds), where nds is the number of Dicke states for N twolevel systems. Filled with ones or the value of degeneracy at each matrix element.

tau_column
(tau, k, j)[source]¶ Determine the column index for the nonzero elements of the matrix for a particular row k and the value of j from the Dicke space.
 Parameters
 tau: str
The tau function to check for this k and j.
 k: int
The row of the matrix M for which the non zero elements have to be calculated.
 j: float
The value of j for this row.
Lattice¶
Lattice Properties¶

cell_structures
(val_s=None, val_t=None, val_u=None)[source]¶ Returns two matrices H_cell and cell_T to help the user form the inputs for defining an instance of Lattice1d and Lattice2d classes. The two matrices are the intra and inter cell Hamiltonians with the tensor structure of the specified site numbers and/or degrees of freedom defined by the user.
 Parameters
 val_slist of str/str
The first list of str’s specifying the sites/degrees of freedom in the unitcell
 val_tlist of str/str
The second list of str’s specifying the sites/degrees of freedom in the unitcell
 val_ulist of str/str
The third list of str’s specifying the sites/degrees of freedom in the unitcell
 Returns
 H_cell_slist of list of str
tensor structure of the cell Hamiltonian elements
 T_inter_cell_slist of list of str
tensor structure of the inter cell Hamiltonian elements
 H_cellQobj
A Qobj initiated with all 0s with proper shape for an input as Hamiltonian_of_cell in Lattice1d.__init__()
 T_inter_cellQobj
A Qobj initiated with all 0s with proper shape for an input as inter_hop in Lattice1d.__init__()
Topology¶

berry_curvature
(eigfs)[source]¶ Computes the discretized Berry curvature on the two dimensional grid of parameters. The function works well for cases with no band mixing.
 Parameters
 eigfsnumpy ndarray
4 dimensional numpy ndarray where the first two indices are for the two discrete values of the two parameters and the third is the index of the occupied bands. The fourth dimension holds the eigenfunctions.
 Returns
 b_curvnumpy ndarray
A two dimensional array of the discretized Berry curvature defined for the values of the two parameters defined in the eigfs.
Visualization¶
Pseudoprobability Functions¶

qfunc
(state, xvec, yvec, g=1.4142135623730951)[source]¶ Qfunction of a given state vector or density matrix at points xvec + i * yvec.
 Parameters
 stateqobj
A state vector or density matrix.
 xvecarray_like
xcoordinates at which to calculate the Wigner function.
 yvecarray_like
ycoordinates at which to calculate the Wigner function.
 gfloat
Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). The value of g is related to the value of hbar in the commutation relation [x, y] = 1j * hbar via hbar=2/g^2 giving the default value hbar=1.
 Returns
 Qarray
Values representing the Qfunction calculated over the specified range [xvec,yvec].

spin_q_function
(rho, theta, phi)[source]¶ Husimi Qfunction for spins.
 Parameters
 stateqobj
A state vector or density matrix for a spinj quantum system.
 thetaarray_like
thetacoordinates at which to calculate the Q function.
 phiarray_like
phicoordinates at which to calculate the Q function.
 Returns
 Q, THETA, PHI2darray
Values representing the spin Q function at the values specified by THETA and PHI.

spin_wigner
(rho, theta, phi)[source]¶ Wigner function for spins on the Bloch sphere.
 Parameters
 stateqobj
A state vector or density matrix for a spinj quantum system.
 thetaarray_like
thetacoordinates at which to calculate the Q function.
 phiarray_like
phicoordinates at which to calculate the Q function.
 Returns
 W, THETA, PHI2darray
Values representing the spin Wigner function at the values specified by THETA and PHI.
Notes
Experimental.

wigner
(psi, xvec, yvec, method='clenshaw', g=1.4142135623730951, sparse=False, parfor=False)[source]¶ Wigner function for a state vector or density matrix at points xvec + i * yvec.
 Parameters
 stateqobj
A state vector or density matrix.
 xvecarray_like
xcoordinates at which to calculate the Wigner function.
 yvecarray_like
ycoordinates at which to calculate the Wigner function. Does not apply to the ‘fft’ method.
 gfloat
Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). The value of g is related to the value of hbar in the commutation relation [x, y] = i * hbar via hbar=2/g^2 giving the default value hbar=1.
 methodstring {‘clenshaw’, ‘iterative’, ‘laguerre’, ‘fft’}
Select method ‘clenshaw’ ‘iterative’, ‘laguerre’, or ‘fft’, where ‘clenshaw’ and ‘iterative’ use an iterative method to evaluate the Wigner functions for density matrices \(m><n\), while ‘laguerre’ uses the Laguerre polynomials in scipy for the same task. The ‘fft’ method evaluates the Fourier transform of the density matrix. The ‘iterative’ method is default, and in general recommended, but the ‘laguerre’ method is more efficient for very sparse density matrices (e.g., superpositions of Fock states in a large Hilbert space). The ‘clenshaw’ method is the preferred method for dealing with density matrices that have a large number of excitations (>~50). ‘clenshaw’ is a fast and numerically stable method.
 sparsebool {False, True}
Tells the default solver whether or not to keep the input density matrix in sparse format. As the dimensions of the density matrix grow, setthing this flag can result in increased performance.
 parforbool {False, True}
Flag for calculating the Laguerre polynomial based Wigner function method=’laguerre’ in parallel using the parfor function.
 Returns
 Warray
Values representing the Wigner function calculated over the specified range [xvec,yvec].
 yvexarray
FFT ONLY. Returns the ycoordinate values calculated via the Fourier transform.
Notes
The ‘fft’ method accepts only an xvec input for the xcoordinate. The ycoordinates are calculated internally.
References
Ulf Leonhardt, Measuring the Quantum State of Light, (Cambridge University Press, 1997)
Graphs and Visualization¶
Functions for visualizing results of quantum dynamics simulations, visualizations of quantum states and processes.

hinton
(rho, xlabels=None, ylabels=None, title=None, ax=None, cmap=None, label_top=True)[source]¶ Draws a Hinton diagram for visualizing a density matrix or superoperator.
 Parameters
 rhoqobj
Input density matrix or superoperator.
 xlabelslist of strings or False
list of x labels
 ylabelslist of strings or False
list of y labels
 titlestring
title of the plot (optional)
 axa matplotlib axes instance
The axes context in which the plot will be drawn.
 cmapa matplotlib colormap instance
Color map to use when plotting.
 label_topbool
If True, xaxis labels will be placed on top, otherwise they will appear below the plot.
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.
 Raises
 ValueError
Input argument is not a quantum object.

matrix_histogram
(M, xlabels=None, ylabels=None, title=None, limits=None, colorbar=True, fig=None, ax=None)[source]¶ Draw a histogram for the matrix M, with the given x and y labels and title.
 Parameters
 MMatrix of Qobj
The matrix to visualize
 xlabelslist of strings
list of x labels
 ylabelslist of strings
list of y labels
 titlestring
title of the plot (optional)
 limitslist/array with two float numbers
The zaxis limits [min, max] (optional)
 axa matplotlib axes instance
The axes context in which the plot will be drawn.
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.
 Raises
 ValueError
Input argument is not valid.

matrix_histogram_complex
(M, xlabels=None, ylabels=None, title=None, limits=None, phase_limits=None, colorbar=True, fig=None, ax=None, threshold=None)[source]¶ Draw a histogram for the amplitudes of matrix M, using the argument of each element for coloring the bars, with the given x and y labels and title.
 Parameters
 MMatrix of Qobj
The matrix to visualize
 xlabelslist of strings
list of x labels
 ylabelslist of strings
list of y labels
 titlestring
title of the plot (optional)
 limitslist/array with two float numbers
The zaxis limits [min, max] (optional)
 phase_limitslist/array with two float numbers
The phaseaxis (colorbar) limits [min, max] (optional)
 axa matplotlib axes instance
The axes context in which the plot will be drawn.
 threshold: float (None)
Threshold for when bars of smaller height should be transparent. If not set, all bars are colored according to the color map.
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.
 Raises
 ValueError
Input argument is not valid.

plot_energy_levels
(H_list, N=0, labels=None, show_ylabels=False, figsize=(8, 12), fig=None, ax=None)[source]¶ Plot the energy level diagrams for a list of Hamiltonians. Include up to N energy levels. For each element in H_list, the energy levels diagram for the cummulative Hamiltonian sum(H_list[0:n]) is plotted, where n is the index of an element in H_list.
 Parameters
 H_listList of Qobj
A list of Hamiltonians.
 labelsList of string
A list of labels for each Hamiltonian
 show_ylabelsBool (default False)
Show y labels to the left of energy levels of the initial Hamiltonian.
 Nint
The number of energy levels to plot
 figsizetuple (int,int)
The size of the figure (width, height).
 figa matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance
The axes context in which the plot will be drawn.
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.
 Raises
 ValueError
Input argument is not valid.

plot_fock_distribution
(rho, offset=0, fig=None, ax=None, figsize=(8, 6), title=None, unit_y_range=True)[source]¶ Plot the Fock distribution for a density matrix (or ket) that describes an oscillator mode.
 Parameters
 rho
qutip.qobj.Qobj
The density matrix (or ket) of the state to visualize.
 figa matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance
The axes context in which the plot will be drawn.
 titlestring
An optional title for the figure.
 figsize(width, height)
The size of the matplotlib figure (in inches) if it is to be created (that is, if no ‘fig’ and ‘ax’ arguments are passed).
 rho
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

plot_wigner_fock_distribution
(rho, fig=None, axes=None, figsize=(8, 4), cmap=None, alpha_max=7.5, colorbar=False, method='iterative', projection='2d')[source]¶ Plot the Fock distribution and the Wigner function for a density matrix (or ket) that describes an oscillator mode.
 Parameters
 rho
qutip.qobj.Qobj
The density matrix (or ket) of the state to visualize.
 figa matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
 axesa list of two matplotlib axes instances
The axes context in which the plot will be drawn.
 figsize(width, height)
The size of the matplotlib figure (in inches) if it is to be created (that is, if no ‘fig’ and ‘ax’ arguments are passed).
 cmapa matplotlib cmap instance
The colormap.
 alpha_maxfloat
The span of the x and y coordinates (both [alpha_max, alpha_max]).
 colorbarbool
Whether (True) or not (False) a colorbar should be attached to the Wigner function graph.
 methodstring {‘iterative’, ‘laguerre’, ‘fft’}
The method used for calculating the wigner function. See the documentation for qutip.wigner for details.
 projection: string {‘2d’, ‘3d’}
Specify whether the Wigner function is to be plotted as a contour graph (‘2d’) or surface plot (‘3d’).
 rho
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

plot_wigner
(rho, fig=None, ax=None, figsize=(6, 6), cmap=None, alpha_max=7.5, colorbar=False, method='clenshaw', projection='2d')[source]¶ Plot the the Wigner function for a density matrix (or ket) that describes an oscillator mode.
 Parameters
 rho
qutip.qobj.Qobj
The density matrix (or ket) of the state to visualize.
 figa matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance
The axes context in which the plot will be drawn.
 figsize(width, height)
The size of the matplotlib figure (in inches) if it is to be created (that is, if no ‘fig’ and ‘ax’ arguments are passed).
 cmapa matplotlib cmap instance
The colormap.
 alpha_maxfloat
The span of the x and y coordinates (both [alpha_max, alpha_max]).
 colorbarbool
Whether (True) or not (False) a colorbar should be attached to the Wigner function graph.
 methodstring {‘clenshaw’, ‘iterative’, ‘laguerre’, ‘fft’}
The method used for calculating the wigner function. See the documentation for qutip.wigner for details.
 projection: string {‘2d’, ‘3d’}
Specify whether the Wigner function is to be plotted as a contour graph (‘2d’) or surface plot (‘3d’).
 rho
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

sphereplot
(theta, phi, values, fig=None, ax=None, save=False)[source]¶ Plots a matrix of values on a sphere
 Parameters
 thetafloat
Angle with respect to zaxis
 phifloat
Angle in xy plane
 valuesarray
Data set to be plotted
 figa matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance
The axes context in which the plot will be drawn.
 savebool {False , True}
Whether to save the figure or not
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

plot_schmidt
(ket, splitting=None, labels_iteration=(3, 2), theme='light', fig=None, ax=None, figsize=(6, 6))[source]¶ Plotting scheme related to Schmidt decomposition. Converts a state into a matrix (A_ij > A_i^j), where rows are first particles and columns  last.
See also: plot_qubism with how=’before_after’ for a similar plot.
 Parameters
 ketQobj
Pure state for plotting.
 splittingint
Plot for a number of first particles versus the rest. If not given, it is (number of particles + 1) // 2.
 theme‘light’ (default) or ‘dark’
Set coloring theme for mapping complex values into colors. See: complex_array_to_rgb.
 labels_iterationint or pair of ints (default (3,2))
Number of particles to be shown as tick labels, for first (vertical) and last (horizontal) particles, respectively.
 figa matplotlib figure instance
The figure canvas on which the plot will be drawn.
 axa matplotlib axis instance
The axis context in which the plot will be drawn.
 figsize(width, height)
The size of the matplotlib figure (in inches) if it is to be created (that is, if no ‘fig’ and ‘ax’ arguments are passed).
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

plot_qubism
(ket, theme='light', how='pairs', grid_iteration=1, legend_iteration=0, fig=None, ax=None, figsize=(6, 6))[source]¶ Qubism plot for pure states of many qudits. Works best for spin chains, especially with even number of particles of the same dimension. Allows to see entanglement between first 2*k particles and the rest.
More information:
J. RodriguezLaguna, P. Migdal, M. Ibanez Berganza, M. Lewenstein, G. Sierra, “Qubism: selfsimilar visualization of manybody wavefunctions”, New J. Phys. 14 053028 (2012), arXiv:1112.3560, http://dx.doi.org/10.1088/13672630/14/5/053028 (open access)
 Parameters
 ketQobj
Pure state for plotting.
 theme‘light’ (default) or ‘dark’
Set coloring theme for mapping complex values into colors. See: complex_array_to_rgb.
 how‘pairs’ (default), ‘pairs_skewed’ or ‘before_after’
Type of Qubism plotting. Options:
‘pairs’  typical coordinates, ‘pairs_skewed’  for ferromagnetic/antriferromagnetic plots, ‘before_after’  related to Schmidt plot (see also: plot_schmidt).
 grid_iterationint (default 1)
Helper lines to be drawn on plot. Show tiles for 2*grid_iteration particles vs all others.
 legend_iterationint (default 0) or ‘grid_iteration’ or ‘all’
Show labels for first 2*legend_iteration particles. Option ‘grid_iteration’ sets the same number of particles
as for grid_iteration.
Option ‘all’ makes label for all particles. Typically it should be 0, 1, 2 or perhaps 3.
 figa matplotlib figure instance
The figure canvas on which the plot will be drawn.
 axa matplotlib axis instance
The axis context in which the plot will be drawn.
 figsize(width, height)
The size of the matplotlib figure (in inches) if it is to be created (that is, if no ‘fig’ and ‘ax’ arguments are passed).
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

plot_expectation_values
(results, ylabels=[], title=None, show_legend=False, fig=None, axes=None, figsize=(8, 4))[source]¶ Visualize the results (expectation values) for an evolution solver. results is assumed to be an instance of Result, or a list of Result instances.
 Parameters
 results(list of)
qutip.solver.Result
List of results objects returned by any of the QuTiP evolution solvers.
 ylabelslist of strings
The yaxis labels. List should be of the same length as results.
 titlestring
The title of the figure.
 show_legendbool
Whether or not to show the legend.
 figa matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
 axesa matplotlib axes instance
The axes context in which the plot will be drawn.
 figsize(width, height)
The size of the matplotlib figure (in inches) if it is to be created (that is, if no ‘fig’ and ‘ax’ arguments are passed).
 results(list of)
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

plot_spin_distribution_2d
(P, THETA, PHI, fig=None, ax=None, figsize=(8, 8))[source]¶ Plot a spin distribution function (given as meshgrid data) with a 2D projection where the surface of the unit sphere is mapped on the unit disk.
 Parameters
 Pmatrix
Distribution values as a meshgrid matrix.
 THETAmatrix
Meshgrid matrix for the theta coordinate.
 PHImatrix
Meshgrid matrix for the phi coordinate.
 figa matplotlib figure instance
The figure canvas on which the plot will be drawn.
 axa matplotlib axis instance
The axis context in which the plot will be drawn.
 figsize(width, height)
The size of the matplotlib figure (in inches) if it is to be created (that is, if no ‘fig’ and ‘ax’ arguments are passed).
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

plot_spin_distribution_3d
(P, THETA, PHI, fig=None, ax=None, figsize=(8, 6))[source]¶ Plots a matrix of values on a sphere
 Parameters
 Pmatrix
Distribution values as a meshgrid matrix.
 THETAmatrix
Meshgrid matrix for the theta coordinate.
 PHImatrix
Meshgrid matrix for the phi coordinate.
 figa matplotlib figure instance
The figure canvas on which the plot will be drawn.
 axa matplotlib axis instance
The axis context in which the plot will be drawn.
 figsize(width, height)
The size of the matplotlib figure (in inches) if it is to be created (that is, if no ‘fig’ and ‘ax’ arguments are passed).
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

plot_wigner_sphere
(fig, ax, wigner, reflections)[source]¶ Plots a coloured Bloch sphere.
 Parameters
 fig
An instance of matplotlib.pyplot.figure.
 ax
An axes instance in fig.
 wignerlist of float
the wigner transformation at steps different theta and phi.
 reflectionsbool
If the reflections of the sphere should be plotted as well.
Notes
Special thanks to Russell P Rundle for writing this function.

sphereplot
(theta, phi, values, fig=None, ax=None, save=False)[source] Plots a matrix of values on a sphere
 Parameters
 thetafloat
Angle with respect to zaxis
 phifloat
Angle in xy plane
 valuesarray
Data set to be plotted
 figa matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance
The axes context in which the plot will be drawn.
 savebool {False , True}
Whether to save the figure or not
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

orbital
(theta, phi, *args)[source]¶ Calculates an angular wave function on a sphere.
psi = orbital(theta,phi,ket1,ket2,...)
calculates the angular wave function on a sphere at the mesh of points defined by theta and phi which is \(\sum_{lm} c_{lm} Y_{lm}(theta,phi)\) where \(C_{lm}\) are the coefficients specified by the list of kets. Each ket has 2l+1 components for some integer l. Parameters
 thetalist/array
Polar angles
 philist/array
Azimuthal angles
 argslist/array
list
of ket vectors.
 Returns
array
for angular wave function
Quantum Process Tomography¶

qpt
(U, op_basis_list)[source]¶ Calculate the quantum process tomography chi matrix for a given (possibly nonunitary) transformation matrix U, which transforms a density matrix in vector form according to:
vec(rho) = U * vec(rho0)
or
rho = vec2mat(U * mat2vec(rho0))
U can be calculated for an open quantum system using the QuTiP propagator function.
 Parameters
 UQobj
Transformation operator. Can be calculated using QuTiP propagator function.
 op_basis_listlist
A list of Qobj’s representing the basis states.
 Returns
 chiarray
QPT chi matrix

qpt_plot
(chi, lbls_list, title=None, fig=None, axes=None)[source]¶ Visualize the quantum process tomography chi matrix. Plot the real and imaginary parts separately.
 Parameters
 chiarray
Input QPT chi matrix.
 lbls_listlist
List of labels for QPT plot axes.
 titlestring
Plot title.
 figfigure instance
User defined figure instance used for generating QPT plot.
 axeslist of figure axis instance
User defined figure axis instance (list of two axes) used for generating QPT plot.
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

qpt_plot_combined
(chi, lbls_list, title=None, fig=None, ax=None, figsize=(8, 6), threshold=None)[source]¶ Visualize the quantum process tomography chi matrix. Plot bars with height and color corresponding to the absolute value and phase, respectively.
 Parameters
 chiarray
Input QPT chi matrix.
 lbls_listlist
List of labels for QPT plot axes.
 titlestring
Plot title.
 figfigure instance
User defined figure instance used for generating QPT plot.
 axfigure axis instance
User defined figure axis instance used for generating QPT plot (alternative to the fig argument).
 threshold: float (None)
Threshold for when bars of smaller height should be transparent. If not set, all bars are colored according to the color map.
 Returns
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.
Quantum Information Processing¶
Gates¶

rx
(phi, N=None, target=0)[source]¶ Singlequbit rotation for operator sigmax with angle phi.
 Returns
 resultqobj
Quantum object for operator describing the rotation.

ry
(phi, N=None, target=0)[source]¶ Singlequbit rotation for operator sigmay with angle phi.
 Returns
 resultqobj
Quantum object for operator describing the rotation.

rz
(phi, N=None, target=0)[source]¶ Singlequbit rotation for operator sigmaz with angle phi.
 Returns
 resultqobj
Quantum object for operator describing the rotation.

sqrtnot
(N=None, target=0)[source]¶ Singlequbit square root NOT gate.
 Returns
 resultqobj
Quantum object for operator describing the square root NOT gate.

snot
(N=None, target=0)[source]¶ Quantum object representing the SNOT (Hadamard) gate.
 Returns
 snot_gateqobj
Quantum object representation of SNOT gate.
Examples
>>> snot() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = True Qobj data = [[ 0.70710678+0.j 0.70710678+0.j] [ 0.70710678+0.j 0.70710678+0.j]]

phasegate
(theta, N=None, target=0)[source]¶ Returns quantum object representing the phase shift gate.
 Parameters
 thetafloat
Phase rotation angle.
 Returns
 phase_gateqobj
Quantum object representation of phase shift gate.
Examples
>>> phasegate(pi/4) Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False Qobj data = [[ 1.00000000+0.j 0.00000000+0.j ] [ 0.00000000+0.j 0.70710678+0.70710678j]]

cphase
(theta, N=2, control=0, target=1)[source]¶ Returns quantum object representing the controlled phase shift gate.
 Parameters
 thetafloat
Phase rotation angle.
 Ninteger
The number of qubits in the target space.
 controlinteger
The index of the control qubit.
 targetinteger
The index of the target qubit.
 Returns
 Uqobj
Quantum object representation of controlled phase gate.

cnot
(N=None, control=0, target=1)[source]¶ Quantum object representing the CNOT gate.
 Returns
 cnot_gateqobj
Quantum object representation of CNOT gate
Examples
>>> cnot() Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = True Qobj data = [[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 1.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 1.+0.j] [ 0.+0.j 0.+0.j 1.+0.j 0.+0.j]]

csign
(N=None, control=0, target=1)[source]¶ Quantum object representing the CSIGN gate.
 Returns
 csign_gateqobj
Quantum object representation of CSIGN gate
Examples
>>> csign() Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = True Qobj data = [[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 1.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 1.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]

berkeley
(N=None, targets=[0, 1])[source]¶ Quantum object representing the Berkeley gate.
 Returns
 berkeley_gateqobj
Quantum object representation of Berkeley gate
Examples
>>> berkeley() Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = True Qobj data = [[ cos(pi/8).+0.j 0.+0.j 0.+0.j 0.+sin(pi/8).j] [ 0.+0.j cos(3pi/8).+0.j 0.+sin(3pi/8).j 0.+0.j] [ 0.+0.j 0.+sin(3pi/8).j cos(3pi/8).+0.j 0.+0.j] [ 0.+sin(pi/8).j 0.+0.j 0.+0.j cos(pi/8).+0.j]]

swapalpha
(alpha, N=None, targets=[0, 1])[source]¶ Quantum object representing the SWAPalpha gate.
 Returns
 swapalpha_gateqobj
Quantum object representation of SWAPalpha gate
Examples
>>> swapalpha(alpha) Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = True Qobj data = [[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.5*(1 + exp(j*pi*alpha) 0.5*(1  exp(j*pi*alpha) 0.+0.j] [ 0.+0.j 0.5*(1  exp(j*pi*alpha) 0.5*(1 + exp(j*pi*alpha) 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]

swap
(N=None, targets=[0, 1])[source]¶ Quantum object representing the SWAP gate.
 Returns
 swap_gateqobj
Quantum object representation of SWAP gate
Examples
>>> swap() Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = True Qobj data = [[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 1.+0.j 0.+0.j] [ 0.+0.j 1.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]

iswap
(N=None, targets=[0, 1])[source]¶ Quantum object representing the iSWAP gate.
 Returns
 iswap_gateqobj
Quantum object representation of iSWAP gate
Examples
>>> iswap() Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = False Qobj data = [[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+1.j 0.+0.j] [ 0.+0.j 0.+1.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]

sqrtswap
(N=None, targets=[0, 1])[source]¶ Quantum object representing the square root SWAP gate.
 Returns
 sqrtswap_gateqobj
Quantum object representation of square root SWAP gate

sqrtiswap
(N=None, targets=[0, 1])[source]¶ Quantum object representing the square root iSWAP gate.
 Returns
 sqrtiswap_gateqobj
Quantum object representation of square root iSWAP gate
Examples
>>> sqrtiswap() Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = False Qobj data = [[ 1.00000000+0.j 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.70710678+0.j 0.000000000.70710678j 0.00000000+0.j] [ 0.00000000+0.j 0.000000000.70710678j 0.70710678+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 1.00000000+0.j]]

fredkin
(N=None, control=0, targets=[1, 2])[source]¶ Quantum object representing the Fredkin gate.
 Returns
 fredkin_gateqobj
Quantum object representation of Fredkin gate.
Examples
>>> fredkin() Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = [8, 8], type = oper, isHerm = True Qobj data = [[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]

toffoli
(N=None, controls=[0, 1], target=2)[source]¶ Quantum object representing the Toffoli gate.
 Returns
 toff_gateqobj
Quantum object representation of Toffoli gate.
Examples
>>> toffoli() Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = [8, 8], type = oper, isHerm = True Qobj data = [[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j] [ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j]]

rotation
(op, phi, N=None, target=0)[source]¶ Singlequbit rotation for operator op with angle phi.
 Returns
 resultqobj
Quantum object for operator describing the rotation.

controlled_gate
(U, N=2, control=0, target=1, control_value=1)[source]¶ Create an Nqubit controlled gate from a singlequbit gate U with the given control and target qubits.
 Parameters
 UQobj
Arbitrary singlequbit gate.
 Ninteger
The number of qubits in the target space.
 controlinteger
The index of the first control qubit.
 targetinteger
The index of the target qubit.
 control_valueinteger (1)
The state of the control qubit that activates the gate U.
 Returns
 resultqobj
Quantum object representing the controlledU gate.

globalphase
(theta, N=1)[source]¶ Returns quantum object representing the global phase shift gate.
 Parameters
 thetafloat
Phase rotation angle.
 Returns
 phase_gateqobj
Quantum object representation of global phase shift gate.
Examples
>>> phasegate(pi/4) Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False Qobj data = [[ 0.70710678+0.70710678j 0.00000000+0.j] [ 0.00000000+0.j 0.70710678+0.70710678j]]

hadamard_transform
(N=1)[source]¶ Quantum object representing the Nqubit Hadamard gate.
 Returns
 qqobj
Quantum object representation of the Nqubit Hadamard gate.

gate_sequence_product
(U_list, left_to_right=True)[source]¶ Calculate the overall unitary matrix for a given list of unitary operations
 Parameters
 U_listlist
List of gates implementing the quantum circuit.
 left_to_rightBoolean
Check if multiplication is to be done from left to right.
 Returns
 U_overallqobj
Overall unitary matrix of a given quantum circuit.

gate_expand_1toN
(U, N, target)[source]¶ Create a Qobj representing a onequbit gate that act on a system with N qubits.
 Parameters
 UQobj
The onequbit gate
 Ninteger
The number of qubits in the target space.
 targetinteger
The index of the target qubit.
 Returns
 gateqobj
Quantum object representation of Nqubit gate.

gate_expand_2toN
(U, N, control=None, target=None, targets=None)[source]¶ Create a Qobj representing a twoqubit gate that act on a system with N qubits.
 Parameters
 UQobj
The twoqubit gate
 Ninteger
The number of qubits in the target space.
 controlinteger
The index of the control qubit.
 targetinteger
The index of the target qubit.
 targetslist
List of target qubits.
 Returns
 gateqobj
Quantum object representation of Nqubit gate.

gate_expand_3toN
(U, N, controls=[0, 1], target=2)[source]¶ Create a Qobj representing a threequbit gate that act on a system with N qubits.
 Parameters
 UQobj
The threequbit gate
 Ninteger
The number of qubits in the target space.
 controlslist
The list of the control qubits.
 targetinteger
The index of the target qubit.
 Returns
 gateqobj
Quantum object representation of Nqubit gate.

expand_operator
(oper, N, targets, dims=None, cyclic_permutation=False)[source]¶ Expand a qubits operator to one that acts on a Nqubit system.
 Parameters
 oper
qutip.Qobj
An operator acts on qubits, the type of the
qutip.Qobj
has to be an operator and the dimension matches the tensored qubit Hilbert space e.g. dims =[[2, 2, 2], [2, 2, 2]]
 Nint
The number of qubits in the system.
 targetsint or list of int
The indices of qubits that are acted on.
 dimslist, optional
A list of integer for the dimension of each composite system. E.g
[2, 2, 2, 2, 2]
for 5 qubits system. If None, qubits system will be the default option. cyclic_permutationboolean, optional
Expand for all cyclic permutation of the targets. E.g. if
N=3
and oper is a 2qubit operator, the result will be a list of three operators, each acting on qubits 0 and 1, 1 and 2, 2 and 0.
 oper
 Returns
 expanded_oper
qutip.Qobj
The expanded qubits operator acting on a system with N qubits.
 expanded_oper
Notes
This is equivalent to gate_expand_1toN, gate_expand_2toN, gate_expand_3toN in
qutip.qip.gate.py
, but works for any dimension.
Qubits¶
Algorithms¶
This module provides the circuit implementation for Quantum Fourier Transform.

qft
(N=1)[source]¶ Quantum Fourier Transform operator on N qubits.
 Parameters
 Nint
Number of qubits.
 Returns
 QFT: qobj
Quantum Fourier transform operator.

qft_steps
(N=1, swapping=True)[source]¶ Quantum Fourier Transform operator on N qubits returning the individual steps as unitary matrices operating from left to right.
 Parameters
 N: int
Number of qubits.
 swap: boolean
Flag indicating sequence of swap gates to be applied at the end or not.
 Returns
 U_step_list: list of qobj
List of Hadamard and controlled rotation gates implementing QFT.

qft_gate_sequence
(N=1, swapping=True)[source]¶ Quantum Fourier Transform operator on N qubits returning the gate sequence.
 Parameters
 N: int
Number of qubits.
 swap: boolean
Flag indicating sequence of swap gates to be applied at the end or not.
 Returns
 qc: instance of QubitCircuit
Gate sequence of Hadamard and controlled rotation gates implementing QFT.
Noisy Devices¶
NonMarkovian Solvers¶
This module contains an implementation of the nonMarkovian transfer tensor method (TTM), introduced in [1].
[1] Javier Cerrillo and Jianshu Cao, Phys. Rev. Lett 112, 110401 (2014)

ttmsolve
(dynmaps, rho0, times, e_ops=[], learningtimes=None, tensors=None, **kwargs)[source]¶ Solve timeevolution using the Transfer Tensor Method, based on a set of precomputed dynamical maps.
 Parameters
 dynmapslist of
qutip.Qobj
List of precomputed dynamical maps (superoperators), or a callback function that returns the superoperator at a given time.
 rho0
qutip.Qobj
Initial density matrix or state vector (ket).
 timesarray_like
list of times \(t_n\) at which to compute \(\rho(t_n)\). Must be uniformily spaced.
 e_opslist of
qutip.Qobj
/ callback function single operator or list of operators for which to evaluate expectation values.
 learningtimesarray_like
list of times \(t_k\) for which we have knowledge of the dynamical maps \(E(t_k)\).
 tensorsarray_like
optional list of precomputed tensors \(T_k\)
 kwargsdictionary
Optional keyword arguments. See
qutip.nonmarkov.ttm.TTMSolverOptions
.
 dynmapslist of
 Returns
 output:
qutip.solver.Result
An instance of the class
qutip.solver.Result
.
 output:
Optimal control¶
Wrapper functions that will manage the creation of the objects, build the configuration, and execute the algorithm required to optimise a set of ctrl pulses for a given (quantum) system. The fidelity error is some measure of distance of the system evolution from the given target evolution in the time allowed for the evolution. The functions minimise this fidelity error wrt the piecewise control amplitudes in the timeslots
There are currently two quantum control pulse optmisations algorithms implemented in this library. There are accessible through the methods in this module. Both the algorithms use the scipy.optimize methods to minimise the fidelity error with respect to to variables that define the pulse.
GRAPE¶
The default algorithm (as it was implemented here first) is GRAPE GRadient Ascent Pulse Engineering [1][2]. It uses a gradient based method such as BFGS to minimise the fidelity error. This makes convergence very quick when an exact gradient can be calculated, but this limits the factors that can taken into account in the fidelity.
CRAB¶
The CRAB [3][4] algorithm was developed at the University of Ulm. In full it is the Chopped RAndom Basis algorithm. The main difference is that it reduces the number of optimisation variables by defining the control pulses by expansions of basis functions, where the variables are the coefficients. Typically a Fourier series is chosen, i.e. the variables are the Fourier coefficients. Therefore it does not need to compute an explicit gradient. By default it uses the NelderMead method for fidelity error minimisation.
References
N Khaneja et. al. Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms. J. Magn. Reson. 172, 296–305 (2005).
Shai Machnes et.al DYNAMO  Dynamic Framework for Quantum Optimal Control arXiv.1011.4874
Doria, P., Calarco, T. & Montangero, S. Optimal Control Technique for ManyBody Quantum Dynamics. Phys. Rev. Lett. 106, 1–4 (2011).
Caneva, T., Calarco, T. & Montangero, S. Chopped randombasis quantum optimization. Phys. Rev. A  At. Mol. Opt. Phys. 84, (2011).

optimize_pulse
(drift, ctrls, initial, target, num_tslots=None, evo_time=None, tau=None, amp_lbound=None, amp_ubound=None, fid_err_targ=1e10, min_grad=1e10, max_iter=500, max_wall_time=180, alg='GRAPE', alg_params=None, optim_params=None, optim_method='DEF', method_params=None, optim_alg=None, max_metric_corr=None, accuracy_factor=None, dyn_type='GEN_MAT', dyn_params=None, prop_type='DEF', prop_params=None, fid_type='DEF', fid_params=None, phase_option=None, fid_err_scale_factor=None, tslot_type='DEF', tslot_params=None, amp_update_mode=None, init_pulse_type='DEF', init_pulse_params=None, pulse_scaling=1.0, pulse_offset=0.0, ramping_pulse_type=None, ramping_pulse_params=None, log_level=0, out_file_ext=None, gen_stats=False)[source]¶ Optimise a control pulse to minimise the fidelity error. The dynamics of the system in any given timeslot are governed by the combined dynamics generator, i.e. the sum of the drift+ctrl_amp[j]*ctrls[j] The control pulse is an [n_ts, n_ctrls)] array of piecewise amplitudes Starting from an intital (typically random) pulse, a multivariable optimisation algorithm attempts to determines the optimal values for the control pulse to minimise the fidelity error The fidelity error is some measure of distance of the system evolution from the given target evolution in the time allowed for the evolution.
 Parameters
 driftQobj or list of Qobj
the underlying dynamics generator of the system can provide list (of length num_tslots) for time dependent drift
 ctrlsList of Qobj or array like [num_tslots, evo_time]
a list of control dynamics generators. These are scaled by the amplitudes to alter the overall dynamics Array like imput can be provided for time dependent control generators
 initialQobj
starting point for the evolution. Typically the identity matrix
 targetQobj
target transformation, e.g. gate or state, for the time evolution
 num_tslotsinteger or None
number of timeslots. None implies that timeslots will be given in the tau array
 evo_timefloat or None
total time for the evolution None implies that timeslots will be given in the tau array
 tauarray[num_tslots] of floats or None
durations for the timeslots. if this is given then num_tslots and evo_time are dervived from it None implies that timeslot durations will be equal and calculated as evo_time/num_tslots
 amp_lboundfloat or list of floats
lower boundaries for the control amplitudes Can be a scalar value applied to all controls or a list of bounds for each control
 amp_uboundfloat or list of floats
upper boundaries for the control amplitudes Can be a scalar value applied to all controls or a list of bounds for each control
 fid_err_targfloat
Fidelity error target. Pulse optimisation will terminate when the fidelity error falls below this value
 mim_gradfloat
Minimum gradient. When the sum of the squares of the gradients wrt to the control amplitudes falls below this value, the optimisation terminates, assuming local minima
 max_iterinteger
Maximum number of iterations of the optimisation algorithm
 max_wall_timefloat
Maximum allowed elapsed time for the optimisation algorithm
 algstring
Algorithm to use in pulse optimisation. Options are:
‘GRAPE’ (default)  GRadient Ascent Pulse Engineering ‘CRAB’  Chopped RAndom Basis
 alg_paramsDictionary
options that are specific to the algorithm see above
 optim_paramsDictionary
The key value pairs are the attribute name and value used to set attribute values Note: attributes are created if they do not exist already, and are overwritten if they do. Note: method_params are applied afterwards and so may override these
 optim_methodstring
a scipy.optimize.minimize method that will be used to optimise the pulse for minimum fidelity error Note that FMIN, FMIN_BFGS & FMIN_L_BFGS_B will all result in calling these specific scipy.optimize methods Note the LBFGSB is equivalent to FMIN_L_BFGS_B for backwards capatibility reasons. Supplying DEF will given alg dependent result:
GRAPE  Default optim_method is FMIN_L_BFGS_B CRAB  Default optim_method is FMIN
 method_paramsdict
Parameters for the optim_method. Note that where there is an attribute of the Optimizer object or the termination_conditions matching the key that attribute. Otherwise, and in some case also, they are assumed to be method_options for the scipy.optimize.minimize method.
 optim_algstring
Deprecated. Use optim_method.
 max_metric_corrinteger
Deprecated. Use method_params instead
 accuracy_factorfloat
Deprecated. Use method_params instead
 dyn_typestring
Dynamics type, i.e. the type of matrix used to describe the dynamics. Options are UNIT, GEN_MAT, SYMPL (see Dynamics classes for details)
 dyn_paramsdict
Parameters for the Dynamics object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 prop_typestring
Propagator type i.e. the method used to calculate the propagtors and propagtor gradient for each timeslot options are DEF, APPROX, DIAG, FRECHET, AUG_MAT DEF will use the default for the specific dyn_type (see PropagatorComputer classes for details)
 prop_paramsdict
Parameters for the PropagatorComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 fid_typestring
Fidelity error (and fidelity error gradient) computation method Options are DEF, UNIT, TRACEDIFF, TD_APPROX DEF will use the default for the specific dyn_type (See FidelityComputer classes for details)
 fid_paramsdict
Parameters for the FidelityComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 phase_optionstring
Deprecated. Pass in fid_params instead.
 fid_err_scale_factorfloat
Deprecated. Use scale_factor key in fid_params instead.
 tslot_typestring
Method for computing the dynamics generators, propagators and evolution in the timeslots. Options: DEF, UPDATE_ALL, DYNAMIC UPDATE_ALL is the only one that currently works (See TimeslotComputer classes for details)
 tslot_paramsdict
Parameters for the TimeslotComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 amp_update_modestring
Deprecated. Use tslot_type instead.
 init_pulse_typestring
type / shape of pulse(s) used to initialise the the control amplitudes. Options (GRAPE) include:
RND, LIN, ZERO, SINE, SQUARE, TRIANGLE, SAW
DEF is RND (see PulseGen classes for details) For the CRAB the this the guess_pulse_type.
 init_pulse_paramsdict
Parameters for the initial / guess pulse generator object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 pulse_scalingfloat
Linear scale factor for generated initial / guess pulses By default initial pulses are generated with amplitudes in the range (1.0, 1.0). These will be scaled by this parameter
 pulse_offsetfloat
Linear offset for the pulse. That is this value will be added to any initial / guess pulses generated.
 ramping_pulse_typestring
Type of pulse used to modulate the control pulse. It’s intended use for a ramping modulation, which is often required in experimental setups. This is only currently implemented in CRAB. GAUSSIAN_EDGE was added for this purpose.
 ramping_pulse_paramsdict
Parameters for the ramping pulse generator object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 log_levelinteger
level of messaging output from the logger. Options are attributes of qutip.logging_utils, in decreasing levels of messaging, are: DEBUG_INTENSE, DEBUG_VERBOSE, DEBUG, INFO, WARN, ERROR, CRITICAL Anything WARN or above is effectively ‘quiet’ execution, assuming everything runs as expected. The default NOTSET implies that the level will be taken from the QuTiP settings file, which by default is WARN
 out_file_extstring or None
files containing the initial and final control pulse amplitudes are saved to the current directory. The default name will be postfixed with this extension Setting this to None will suppress the output of files
 gen_statsboolean
if set to True then statistics for the optimisation run will be generated  accessible through attributes of the stats object
 Returns
 optOptimResult
Returns instance of OptimResult, which has attributes giving the reason for termination, final fidelity error, final evolution final amplitudes, statistics etc

optimize_pulse_unitary
(H_d, H_c, U_0, U_targ, num_tslots=None, evo_time=None, tau=None, amp_lbound=None, amp_ubound=None, fid_err_targ=1e10, min_grad=1e10, max_iter=500, max_wall_time=180, alg='GRAPE', alg_params=None, optim_params=None, optim_method='DEF', method_params=None, optim_alg=None, max_metric_corr=None, accuracy_factor=None, phase_option='PSU', dyn_params=None, prop_params=None, fid_params=None, tslot_type='DEF', tslot_params=None, amp_update_mode=None, init_pulse_type='DEF', init_pulse_params=None, pulse_scaling=1.0, pulse_offset=0.0, ramping_pulse_type=None, ramping_pulse_params=None, log_level=0, out_file_ext=None, gen_stats=False)[source]¶ Optimise a control pulse to minimise the fidelity error, assuming that the dynamics of the system are generated by unitary operators. This function is simply a wrapper for optimize_pulse, where the appropriate options for unitary dynamics are chosen and the parameter names are in the format familiar to unitary dynamics The dynamics of the system in any given timeslot are governed by the combined Hamiltonian, i.e. the sum of the H_d + ctrl_amp[j]*H_c[j] The control pulse is an [n_ts, n_ctrls] array of piecewise amplitudes Starting from an intital (typically random) pulse, a multivariable optimisation algorithm attempts to determines the optimal values for the control pulse to minimise the fidelity error The maximum fidelity for a unitary system is 1, i.e. when the time evolution resulting from the pulse is equivalent to the target. And therefore the fidelity error is 1  fidelity
 Parameters
 H_dQobj or list of Qobj
Drift (aka system) the underlying Hamiltonian of the system can provide list (of length num_tslots) for time dependent drift
 H_cList of Qobj or array like [num_tslots, evo_time]
a list of control Hamiltonians. These are scaled by the amplitudes to alter the overall dynamics Array like imput can be provided for time dependent control generators
 U_0Qobj
starting point for the evolution. Typically the identity matrix
 U_targQobj
target transformation, e.g. gate or state, for the time evolution
 num_tslotsinteger or None
number of timeslots. None implies that timeslots will be given in the tau array
 evo_timefloat or None
total time for the evolution None implies that timeslots will be given in the tau array
 tauarray[num_tslots] of floats or None
durations for the timeslots. if this is given then num_tslots and evo_time are dervived from it None implies that timeslot durations will be equal and calculated as evo_time/num_tslots
 amp_lboundfloat or list of floats
lower boundaries for the control amplitudes Can be a scalar value applied to all controls or a list of bounds for each control
 amp_uboundfloat or list of floats
upper boundaries for the control amplitudes Can be a scalar value applied to all controls or a list of bounds for each control
 fid_err_targfloat
Fidelity error target. Pulse optimisation will terminate when the fidelity error falls below this value
 mim_gradfloat
Minimum gradient. When the sum of the squares of the gradients wrt to the control amplitudes falls below this value, the optimisation terminates, assuming local minima
 max_iterinteger
Maximum number of iterations of the optimisation algorithm
 max_wall_timefloat
Maximum allowed elapsed time for the optimisation algorithm
 algstring
Algorithm to use in pulse optimisation. Options are:
‘GRAPE’ (default)  GRadient Ascent Pulse Engineering ‘CRAB’  Chopped RAndom Basis
 alg_paramsDictionary
options that are specific to the algorithm see above
 optim_paramsDictionary
The key value pairs are the attribute name and value used to set attribute values Note: attributes are created if they do not exist already, and are overwritten if they do. Note: method_params are applied afterwards and so may override these
 optim_methodstring
a scipy.optimize.minimize method that will be used to optimise the pulse for minimum fidelity error Note that FMIN, FMIN_BFGS & FMIN_L_BFGS_B will all result in calling these specific scipy.optimize methods Note the LBFGSB is equivalent to FMIN_L_BFGS_B for backwards capatibility reasons. Supplying DEF will given alg dependent result:
GRAPE  Default optim_method is FMIN_L_BFGS_B CRAB  Default optim_method is FMIN
 method_paramsdict
Parameters for the optim_method. Note that where there is an attribute of the Optimizer object or the termination_conditions matching the key that attribute. Otherwise, and in some case also, they are assumed to be method_options for the scipy.optimize.minimize method.
 optim_algstring
Deprecated. Use optim_method.
 max_metric_corrinteger
Deprecated. Use method_params instead
 accuracy_factorfloat
Deprecated. Use method_params instead
 phase_optionstring
determines how global phase is treated in fidelity calculations (fid_type=’UNIT’ only). Options:
PSU  global phase ignored SU  global phase included
 dyn_paramsdict
Parameters for the Dynamics object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 prop_paramsdict
Parameters for the PropagatorComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 fid_paramsdict
Parameters for the FidelityComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 tslot_typestring
Method for computing the dynamics generators, propagators and evolution in the timeslots. Options: DEF, UPDATE_ALL, DYNAMIC UPDATE_ALL is the only one that currently works (See TimeslotComputer classes for details)
 tslot_paramsdict
Parameters for the TimeslotComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 amp_update_modestring
Deprecated. Use tslot_type instead.
 init_pulse_typestring
type / shape of pulse(s) used to initialise the the control amplitudes. Options (GRAPE) include:
RND, LIN, ZERO, SINE, SQUARE, TRIANGLE, SAW DEF is RND
(see PulseGen classes for details) For the CRAB the this the guess_pulse_type.
 init_pulse_paramsdict
Parameters for the initial / guess pulse generator object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 pulse_scalingfloat
Linear scale factor for generated initial / guess pulses By default initial pulses are generated with amplitudes in the range (1.0, 1.0). These will be scaled by this parameter
 pulse_offsetfloat
Linear offset for the pulse. That is this value will be added to any initial / guess pulses generated.
 ramping_pulse_typestring
Type of pulse used to modulate the control pulse. It’s intended use for a ramping modulation, which is often required in experimental setups. This is only currently implemented in CRAB. GAUSSIAN_EDGE was added for this purpose.
 ramping_pulse_paramsdict
Parameters for the ramping pulse generator object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 log_levelinteger
level of messaging output from the logger. Options are attributes of qutip.logging_utils, in decreasing levels of messaging, are: DEBUG_INTENSE, DEBUG_VERBOSE, DEBUG, INFO, WARN, ERROR, CRITICAL Anything WARN or above is effectively ‘quiet’ execution, assuming everything runs as expected. The default NOTSET implies that the level will be taken from the QuTiP settings file, which by default is WARN
 out_file_extstring or None
files containing the initial and final control pulse amplitudes are saved to the current directory. The default name will be postfixed with this extension Setting this to None will suppress the output of files
 gen_statsboolean
if set to True then statistics for the optimisation run will be generated  accessible through attributes of the stats object
 Returns
 optOptimResult
Returns instance of OptimResult, which has attributes giving the reason for termination, final fidelity error, final evolution final amplitudes, statistics etc

create_pulse_optimizer
(drift, ctrls, initial, target, num_tslots=None, evo_time=None, tau=None, amp_lbound=None, amp_ubound=None, fid_err_targ=1e10, min_grad=1e10, max_iter=500, max_wall_time=180, alg='GRAPE', alg_params=None, optim_params=None, optim_method='DEF', method_params=None, optim_alg=None, max_metric_corr=None, accuracy_factor=None, dyn_type='GEN_MAT', dyn_params=None, prop_type='DEF', prop_params=None, fid_type='DEF', fid_params=None, phase_option=None, fid_err_scale_factor=None, tslot_type='DEF', tslot_params=None, amp_update_mode=None, init_pulse_type='DEF', init_pulse_params=None, pulse_scaling=1.0, pulse_offset=0.0, ramping_pulse_type=None, ramping_pulse_params=None, log_level=0, gen_stats=False)[source]¶ Generate the objects of the appropriate subclasses required for the pulse optmisation based on the parameters given Note this method may be preferable to calling optimize_pulse if more detailed configuration is required before running the optmisation algorthim, or the algorithm will be run many times, for instances when trying to finding global the optimum or minimum time optimisation
 Parameters
 driftQobj or list of Qobj
the underlying dynamics generator of the system can provide list (of length num_tslots) for time dependent drift
 ctrlsList of Qobj or array like [num_tslots, evo_time]
a list of control dynamics generators. These are scaled by the amplitudes to alter the overall dynamics Array like imput can be provided for time dependent control generators
 initialQobj
starting point for the evolution. Typically the identity matrix
 targetQobj
target transformation, e.g. gate or state, for the time evolution
 num_tslotsinteger or None
number of timeslots. None implies that timeslots will be given in the tau array
 evo_timefloat or None
total time for the evolution None implies that timeslots will be given in the tau array
 tauarray[num_tslots] of floats or None
durations for the timeslots. if this is given then num_tslots and evo_time are dervived from it None implies that timeslot durations will be equal and calculated as evo_time/num_tslots
 amp_lboundfloat or list of floats
lower boundaries for the control amplitudes Can be a scalar value applied to all controls or a list of bounds for each control
 amp_uboundfloat or list of floats
upper boundaries for the control amplitudes Can be a scalar value applied to all controls or a list of bounds for each control
 fid_err_targfloat
Fidelity error target. Pulse optimisation will terminate when the fidelity error falls below this value
 mim_gradfloat
Minimum gradient. When the sum of the squares of the gradients wrt to the control amplitudes falls below this value, the optimisation terminates, assuming local minima
 max_iterinteger
Maximum number of iterations of the optimisation algorithm
 max_wall_timefloat
Maximum allowed elapsed time for the optimisation algorithm
 algstring
Algorithm to use in pulse optimisation. Options are:
‘GRAPE’ (default)  GRadient Ascent Pulse Engineering ‘CRAB’  Chopped RAndom Basis
 alg_paramsDictionary
options that are specific to the algorithm see above
 optim_paramsDictionary
The key value pairs are the attribute name and value used to set attribute values Note: attributes are created if they do not exist already, and are overwritten if they do. Note: method_params are applied afterwards and so may override these
 optim_methodstring
a scipy.optimize.minimize method that will be used to optimise the pulse for minimum fidelity error Note that FMIN, FMIN_BFGS & FMIN_L_BFGS_B will all result in calling these specific scipy.optimize methods Note the LBFGSB is equivalent to FMIN_L_BFGS_B for backwards capatibility reasons. Supplying DEF will given alg dependent result:
GRAPE  Default optim_method is FMIN_L_BFGS_B
CRAB  Default optim_method is NelderMead
 method_paramsdict
Parameters for the optim_method. Note that where there is an attribute of the Optimizer object or the termination_conditions matching the key that attribute. Otherwise, and in some case also, they are assumed to be method_options for the scipy.optimize.minimize method.
 optim_algstring
Deprecated. Use optim_method.
 max_metric_corrinteger
Deprecated. Use method_params instead
 accuracy_factorfloat
Deprecated. Use method_params instead
 dyn_typestring
Dynamics type, i.e. the type of matrix used to describe the dynamics. Options are UNIT, GEN_MAT, SYMPL (see Dynamics classes for details)
 dyn_paramsdict
Parameters for the Dynamics object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 prop_typestring
Propagator type i.e. the method used to calculate the propagtors and propagtor gradient for each timeslot options are DEF, APPROX, DIAG, FRECHET, AUG_MAT DEF will use the default for the specific dyn_type (see PropagatorComputer classes for details)
 prop_paramsdict
Parameters for the PropagatorComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 fid_typestring
Fidelity error (and fidelity error gradient) computation method Options are DEF, UNIT, TRACEDIFF, TD_APPROX DEF will use the default for the specific dyn_type (See FidelityComputer classes for details)
 fid_paramsdict
Parameters for the FidelityComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 phase_optionstring
Deprecated. Pass in fid_params instead.
 fid_err_scale_factorfloat
Deprecated. Use scale_factor key in fid_params instead.
 tslot_typestring
Method for computing the dynamics generators, propagators and evolution in the timeslots. Options: DEF, UPDATE_ALL, DYNAMIC UPDATE_ALL is the only one that currently works (See TimeslotComputer classes for details)
 tslot_paramsdict
Parameters for the TimeslotComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 amp_update_modestring
Deprecated. Use tslot_type instead.
 init_pulse_typestring
type / shape of pulse(s) used to initialise the the control amplitudes. Options (GRAPE) include:
RND, LIN, ZERO, SINE, SQUARE, TRIANGLE, SAW DEF is RND
(see PulseGen classes for details) For the CRAB the this the guess_pulse_type.
 init_pulse_paramsdict
Parameters for the initial / guess pulse generator object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 pulse_scalingfloat
Linear scale factor for generated initial / guess pulses By default initial pulses are generated with amplitudes in the range (1.0, 1.0). These will be scaled by this parameter
 pulse_offsetfloat
Linear offset for the pulse. That is this value will be added to any initial / guess pulses generated.
 ramping_pulse_typestring
Type of pulse used to modulate the control pulse. It’s intended use for a ramping modulation, which is often required in experimental setups. This is only currently implemented in CRAB. GAUSSIAN_EDGE was added for this purpose.
 ramping_pulse_paramsdict
Parameters for the ramping pulse generator object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 log_levelinteger
level of messaging output from the logger. Options are attributes of qutip.logging_utils, in decreasing levels of messaging, are: DEBUG_INTENSE, DEBUG_VERBOSE, DEBUG, INFO, WARN, ERROR, CRITICAL Anything WARN or above is effectively ‘quiet’ execution, assuming everything runs as expected. The default NOTSET implies that the level will be taken from the QuTiP settings file, which by default is WARN
 gen_statsboolean
if set to True then statistics for the optimisation run will be generated  accessible through attributes of the stats object
 Returns
 optOptimizer
Instance of an Optimizer, through which the Config, Dynamics, PulseGen, and TerminationConditions objects can be accessed as attributes. The PropagatorComputer, FidelityComputer and TimeslotComputer objects can be accessed as attributes of the Dynamics object, e.g. optimizer.dynamics.fid_computer The optimisation can be run through the optimizer.run_optimization

opt_pulse_crab
(drift, ctrls, initial, target, num_tslots=None, evo_time=None, tau=None, amp_lbound=None, amp_ubound=None, fid_err_targ=1e05, max_iter=500, max_wall_time=180, alg_params=None, num_coeffs=None, init_coeff_scaling=1.0, optim_params=None, optim_method='fmin', method_params=None, dyn_type='GEN_MAT', dyn_params=None, prop_type='DEF', prop_params=None, fid_type='DEF', fid_params=None, tslot_type='DEF', tslot_params=None, guess_pulse_type=None, guess_pulse_params=None, guess_pulse_scaling=1.0, guess_pulse_offset=0.0, guess_pulse_action='MODULATE', ramping_pulse_type=None, ramping_pulse_params=None, log_level=0, out_file_ext=None, gen_stats=False)[source]¶ Optimise a control pulse to minimise the fidelity error. The dynamics of the system in any given timeslot are governed by the combined dynamics generator, i.e. the sum of the drift+ctrl_amp[j]*ctrls[j] The control pulse is an [n_ts, n_ctrls] array of piecewise amplitudes. The CRAB algorithm uses basis function coefficents as the variables to optimise. It does NOT use any gradient function. A multivariable optimisation algorithm attempts to determines the optimal values for the control pulse to minimise the fidelity error The fidelity error is some measure of distance of the system evolution from the given target evolution in the time allowed for the evolution.
 Parameters
 driftQobj or list of Qobj
the underlying dynamics generator of the system can provide list (of length num_tslots) for time dependent drift
 ctrlsList of Qobj or array like [num_tslots, evo_time]
a list of control dynamics generators. These are scaled by the amplitudes to alter the overall dynamics Array like imput can be provided for time dependent control generators
 initialQobj
starting point for the evolution. Typically the identity matrix
 targetQobj
target transformation, e.g. gate or state, for the time evolution
 num_tslotsinteger or None
number of timeslots. None implies that timeslots will be given in the tau array
 evo_timefloat or None
total time for the evolution None implies that timeslots will be given in the tau array
 tauarray[num_tslots] of floats or None
durations for the timeslots. if this is given then num_tslots and evo_time are dervived from it None implies that timeslot durations will be equal and calculated as evo_time/num_tslots
 amp_lboundfloat or list of floats
lower boundaries for the control amplitudes Can be a scalar value applied to all controls or a list of bounds for each control
 amp_uboundfloat or list of floats
upper boundaries for the control amplitudes Can be a scalar value applied to all controls or a list of bounds for each control
 fid_err_targfloat
Fidelity error target. Pulse optimisation will terminate when the fidelity error falls below this value
 max_iterinteger
Maximum number of iterations of the optimisation algorithm
 max_wall_timefloat
Maximum allowed elapsed time for the optimisation algorithm
 alg_paramsDictionary
options that are specific to the algorithm see above
 optim_paramsDictionary
The key value pairs are the attribute name and value used to set attribute values Note: attributes are created if they do not exist already, and are overwritten if they do. Note: method_params are applied afterwards and so may override these
 coeff_scalingfloat
Linear scale factor for the random basis coefficients By default these range from 1.0 to 1.0 Note this is overridden by alg_params (if given there)
 num_coeffsinteger
Number of coefficients used for each basis function Note this is calculated automatically based on the dimension of the dynamics if not given. It is crucial to the performane of the algorithm that it is set as low as possible, while still giving high enough frequencies. Note this is overridden by alg_params (if given there)
 optim_methodstring
Multivariable optimisation method The only tested options are ‘fmin’ and ‘Neldermead’ In theory any nongradient method implemented in scipy.optimize.mininize could be used.
 method_paramsdict
Parameters for the optim_method. Note that where there is an attribute of the Optimizer object or the termination_conditions matching the key that attribute. Otherwise, and in some case also, they are assumed to be method_options for the scipy.optimize.minimize method. The commonly used parameter are:
xtol  limit on variable change for convergence ftol  limit on fidelity error change for convergence
 dyn_typestring
Dynamics type, i.e. the type of matrix used to describe the dynamics. Options are UNIT, GEN_MAT, SYMPL (see Dynamics classes for details)
 dyn_paramsdict
Parameters for the Dynamics object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 prop_typestring
Propagator type i.e. the method used to calculate the propagtors and propagtor gradient for each timeslot options are DEF, APPROX, DIAG, FRECHET, AUG_MAT DEF will use the default for the specific dyn_type (see PropagatorComputer classes for details)
 prop_paramsdict
Parameters for the PropagatorComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 fid_typestring
Fidelity error (and fidelity error gradient) computation method Options are DEF, UNIT, TRACEDIFF, TD_APPROX DEF will use the default for the specific dyn_type (See FidelityComputer classes for details)
 fid_paramsdict
Parameters for the FidelityComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 tslot_typestring
Method for computing the dynamics generators, propagators and evolution in the timeslots. Options: DEF, UPDATE_ALL, DYNAMIC UPDATE_ALL is the only one that currently works (See TimeslotComputer classes for details)
 tslot_paramsdict
Parameters for the TimeslotComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 guess_pulse_typestring
type / shape of pulse(s) used modulate the control amplitudes. Options include:
RND, LIN, ZERO, SINE, SQUARE, TRIANGLE, SAW, GAUSSIAN
Default is None
 guess_pulse_paramsdict
Parameters for the guess pulse generator object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 guess_pulse_actionstring
Determines how the guess pulse is applied to the pulse generated by the basis expansion. Options are: MODULATE, ADD Default is MODULATE
 pulse_scalingfloat
Linear scale factor for generated guess pulses By default initial pulses are generated with amplitudes in the range (1.0, 1.0). These will be scaled by this parameter
 pulse_offsetfloat
Linear offset for the pulse. That is this value will be added to any guess pulses generated.
 ramping_pulse_typestring
Type of pulse used to modulate the control pulse. It’s intended use for a ramping modulation, which is often required in experimental setups. This is only currently implemented in CRAB. GAUSSIAN_EDGE was added for this purpose.
 ramping_pulse_paramsdict
Parameters for the ramping pulse generator object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 log_levelinteger
level of messaging output from the logger. Options are attributes of qutip.logging_utils, in decreasing levels of messaging, are: DEBUG_INTENSE, DEBUG_VERBOSE, DEBUG, INFO, WARN, ERROR, CRITICAL Anything WARN or above is effectively ‘quiet’ execution, assuming everything runs as expected. The default NOTSET implies that the level will be taken from the QuTiP settings file, which by default is WARN
 out_file_extstring or None
files containing the initial and final control pulse amplitudes are saved to the current directory. The default name will be postfixed with this extension Setting this to None will suppress the output of files
 gen_statsboolean
if set to True then statistics for the optimisation run will be generated  accessible through attributes of the stats object
 Returns
 optOptimResult
Returns instance of OptimResult, which has attributes giving the reason for termination, final fidelity error, final evolution final amplitudes, statistics etc

opt_pulse_crab_unitary
(H_d, H_c, U_0, U_targ, num_tslots=None, evo_time=None, tau=None, amp_lbound=None, amp_ubound=None, fid_err_targ=1e05, max_iter=500, max_wall_time=180, alg_params=None, num_coeffs=None, init_coeff_scaling=1.0, optim_params=None, optim_method='fmin', method_params=None, phase_option='PSU', dyn_params=None, prop_params=None, fid_params=None, tslot_type='DEF', tslot_params=None, guess_pulse_type=None, guess_pulse_params=None, guess_pulse_scaling=1.0, guess_pulse_offset=0.0, guess_pulse_action='MODULATE', ramping_pulse_type=None, ramping_pulse_params=None, log_level=0, out_file_ext=None, gen_stats=False)[source]¶ Optimise a control pulse to minimise the fidelity error, assuming that the dynamics of the system are generated by unitary operators. This function is simply a wrapper for optimize_pulse, where the appropriate options for unitary dynamics are chosen and the parameter names are in the format familiar to unitary dynamics The dynamics of the system in any given timeslot are governed by the combined Hamiltonian, i.e. the sum of the H_d + ctrl_amp[j]*H_c[j] The control pulse is an [n_ts, n_ctrls] array of piecewise amplitudes
The CRAB algorithm uses basis function coefficents as the variables to optimise. It does NOT use any gradient function. A multivariable optimisation algorithm attempts to determines the optimal values for the control pulse to minimise the fidelity error The fidelity error is some measure of distance of the system evolution from the given target evolution in the time allowed for the evolution.
 Parameters
 H_dQobj or list of Qobj
Drift (aka system) the underlying Hamiltonian of the system can provide list (of length num_tslots) for time dependent drift
 H_cList of Qobj or array like [num_tslots, evo_time]
a list of control Hamiltonians. These are scaled by the amplitudes to alter the overall dynamics Array like imput can be provided for time dependent control generators
 U_0Qobj
starting point for the evolution. Typically the identity matrix
 U_targQobj
target transformation, e.g. gate or state, for the time evolution
 num_tslotsinteger or None
number of timeslots. None implies that timeslots will be given in the tau array
 evo_timefloat or None
total time for the evolution None implies that timeslots will be given in the tau array
 tauarray[num_tslots] of floats or None
durations for the timeslots. if this is given then num_tslots and evo_time are dervived from it None implies that timeslot durations will be equal and calculated as evo_time/num_tslots
 amp_lboundfloat or list of floats
lower boundaries for the control amplitudes Can be a scalar value applied to all controls or a list of bounds for each control
 amp_uboundfloat or list of floats
upper boundaries for the control amplitudes Can be a scalar value applied to all controls or a list of bounds for each control
 fid_err_targfloat
Fidelity error target. Pulse optimisation will terminate when the fidelity error falls below this value
 max_iterinteger
Maximum number of iterations of the optimisation algorithm
 max_wall_timefloat
Maximum allowed elapsed time for the optimisation algorithm
 alg_paramsDictionary
options that are specific to the algorithm see above
 optim_paramsDictionary
The key value pairs are the attribute name and value used to set attribute values Note: attributes are created if they do not exist already, and are overwritten if they do. Note: method_params are applied afterwards and so may override these
 coeff_scalingfloat
Linear scale factor for the random basis coefficients By default these range from 1.0 to 1.0 Note this is overridden by alg_params (if given there)
 num_coeffsinteger
Number of coefficients used for each basis function Note this is calculated automatically based on the dimension of the dynamics if not given. It is crucial to the performane of the algorithm that it is set as low as possible, while still giving high enough frequencies. Note this is overridden by alg_params (if given there)
 optim_methodstring
Multivariable optimisation method The only tested options are ‘fmin’ and ‘Neldermead’ In theory any nongradient method implemented in scipy.optimize.mininize could be used.
 method_paramsdict
Parameters for the optim_method. Note that where there is an attribute of the Optimizer object or the termination_conditions matching the key that attribute. Otherwise, and in some case also, they are assumed to be method_options for the scipy.optimize.minimize method. The commonly used parameter are:
xtol  limit on variable change for convergence ftol  limit on fidelity error change for convergence
 phase_optionstring
determines how global phase is treated in fidelity calculations (fid_type=’UNIT’ only). Options:
PSU  global phase ignored SU  global phase included
 dyn_paramsdict
Parameters for the Dynamics object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 prop_paramsdict
Parameters for the PropagatorComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 fid_paramsdict
Parameters for the FidelityComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 tslot_typestring
Method for computing the dynamics generators, propagators and evolution in the timeslots. Options: DEF, UPDATE_ALL, DYNAMIC UPDATE_ALL is the only one that currently works (See TimeslotComputer classes for details)
 tslot_paramsdict
Parameters for the TimeslotComputer object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 guess_pulse_typestring
type / shape of pulse(s) used modulate the control amplitudes. Options include:
RND, LIN, ZERO, SINE, SQUARE, TRIANGLE, SAW, GAUSSIAN
Default is None
 guess_pulse_paramsdict
Parameters for the guess pulse generator object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 guess_pulse_actionstring
Determines how the guess pulse is applied to the pulse generated by the basis expansion. Options are: MODULATE, ADD Default is MODULATE
 pulse_scalingfloat
Linear scale factor for generated guess pulses By default initial pulses are generated with amplitudes in the range (1.0, 1.0). These will be scaled by this parameter
 pulse_offsetfloat
Linear offset for the pulse. That is this value will be added to any guess pulses generated.
 ramping_pulse_typestring
Type of pulse used to modulate the control pulse. It’s intended use for a ramping modulation, which is often required in experimental setups. This is only currently implemented in CRAB. GAUSSIAN_EDGE was added for this purpose.
 ramping_pulse_paramsdict
Parameters for the ramping pulse generator object The key value pairs are assumed to be attribute name value pairs They applied after the object is created
 log_levelinteger
level of messaging output from the logger. Options are attributes of qutip.logging_utils, in decreasing levels of messaging, are: DEBUG_INTENSE, DEBUG_VERBOSE, DEBUG, INFO, WARN, ERROR, CRITICAL Anything WARN or above is effectively ‘quiet’ execution, assuming everything runs as expected. The default NOTSET implies that the level will be taken from the QuTiP settings file, which by default is WARN
 out_file_extstring or None
files containing the initial and final control pulse amplitudes are saved to the current directory. The default name will be postfixed with this extension Setting this to None will suppress the output of files
 gen_statsboolean
if set to True then statistics for the optimisation run will be generated  accessible through attributes of the stats object
 Returns
 optOptimResult
Returns instance of OptimResult, which has attributes giving the reason for termination, final fidelity error, final evolution final amplitudes, statistics etc
Pulse generator  Generate pulses for the timeslots Each class defines a gen_pulse function that produces a float array of size num_tslots. Each class produces a differ type of pulse. See the class and gen_pulse function descriptions for details

create_pulse_gen
(pulse_type='RND', dyn=None, pulse_params=None)[source]¶ Create and return a pulse generator object matching the given type. The pulse generators each produce a different type of pulse, see the gen_pulse function description for details. These are the random pulse options:
RND  Independent random value in each timeslot RNDFOURIER  Fourier series with random coefficients RNDWAVES  Summation of random waves RNDWALK1  Random change in amplitude each timeslot RNDWALK2  Random change in amp gradient each timeslot
These are the other nonperiodic options:
LIN  Linear, i.e. contant gradient over the time ZERO  special case of the LIN pulse, where the gradient is 0
These are the periodic options
SINE  Sine wave SQUARE  Square wave SAW  Saw tooth wave TRIANGLE  Triangular wave
If a Dynamics object is passed in then this is used in instantiate the PulseGen, meaning that some timeslot and amplitude properties are copied over.
Utility Functions¶
Graph Theory Routines¶
This module contains a collection of graph theory routines used mainly to reorder matrices for iterative steady state solvers.

breadth_first_search
(A, start)[source]¶ BreadthFirstSearch (BFS) of a graph in CSR or CSC matrix format starting from a given node (row). Takes Qobjs and CSR or CSC matrices as inputs.
This function requires a matrix with symmetric structure. Use A+trans(A) if original matrix is not symmetric or not sure.
 Parameters
 Acsc_matrix, csr_matrix
Input graph in CSC or CSR matrix format
 startint
Staring node for BFS traversal.
 Returns
 orderarray
Order in which nodes are traversed from starting node.
 levelsarray
Level of the nodes in the order that they are traversed.

graph_degree
(A)[source]¶ Returns the degree for the nodes (rows) of a symmetric graph in sparse CSR or CSC format, or a qobj.
 Parameters
 Aqobj, csr_matrix, csc_matrix
Input quantum object or csr_matrix.
 Returns
 degreearray
Array of integers giving the degree for each node (row).

reverse_cuthill_mckee
(A, sym=False)[source]¶ Returns the permutation array that orders a sparse CSR or CSC matrix in ReverseCuthill McKee ordering. Since the input matrix must be symmetric, this routine works on the matrix A+Trans(A) if the sym flag is set to False (Default).
It is assumed by default (sym=False) that the input matrix is not symmetric. This is because it is faster to do A+Trans(A) than it is to check for symmetry for a generic matrix. If you are guaranteed that the matrix is symmetric in structure (values of matrix element do not matter) then set sym=True
 Parameters
 Acsc_matrix, csr_matrix
Input sparse CSC or CSR sparse matrix format.
 symbool {False, True}
Flag to set whether input matrix is symmetric.
 Returns
 permarray
Array of permuted row and column indices.
Notes
This routine is used primarily for internal reordering of Lindblad superoperators for use in iterative solver routines.
References
E. Cuthill and J. McKee, “Reducing the Bandwidth of Sparse Symmetric Matrices”, ACM ‘69 Proceedings of the 1969 24th national conference, (1969).

maximum_bipartite_matching
(A, perm_type='row')[source]¶ Returns an array of row or column permutations that removes nonzero elements from the diagonal of a nonsingular square CSC sparse matrix. Such a permutation is always possible provided that the matrix is nonsingular. This function looks at the structure of the matrix only.
The input matrix will be converted to CSC matrix format if necessary.
 Parameters
 Asparse matrix
Input matrix
 perm_typestr {‘row’, ‘column’}
Type of permutation to generate.
 Returns
 permarray
Array of row or column permutations.
Notes
This function relies on a maximum cardinality bipartite matching algorithm based on a breadthfirst search (BFS) of the underlying graph[1]_.
References
I. S. Duff, K. Kaya, and B. Ucar, “Design, Implementation, and Analysis of Maximum Transversal Algorithms”, ACM Trans. Math. Softw. 38, no. 2, (2011).

weighted_bipartite_matching
(A, perm_type='row')[source]¶ Returns an array of row permutations that attempts to maximize the product of the ABS values of the diagonal elements in a nonsingular square CSC sparse matrix. Such a permutation is always possible provided that the matrix is nonsingular.
This function looks at both the structure and ABS values of the underlying matrix.
 Parameters
 Acsc_matrix
Input matrix
 perm_typestr {‘row’, ‘column’}
Type of permutation to generate.
 Returns
 permarray
Array of row or column permutations.
Notes
This function uses a weighted maximum cardinality bipartite matching algorithm based on breadthfirst search (BFS). The columns are weighted according to the element of max ABS value in the associated rows and are traversed in descending order by weight. When performing the BFS traversal, the row associated to a given column is the one with maximum weight. Unlike other techniques[1]_, this algorithm does not guarantee the product of the diagonal is maximized. However, this limitation is offset by the substantially faster runtime of this method.
References
I. S. Duff and J. Koster, “The design and use of algorithms for permuting large entries to the diagonal of sparse matrices”, SIAM J. Matrix Anal. and Applics. 20, no. 4, 889 (1997).
Utility Functions¶
This module contains utility functions that are commonly needed in other qutip modules.

n_thermal
(w, w_th)[source]¶ Return the number of photons in thermal equilibrium for an harmonic oscillator mode with frequency ‘w’, at the temperature described by ‘w_th’ where \(\omega_{\rm th} = k_BT/\hbar\).
 Parameters
 wfloat or array
Frequency of the oscillator.
 w_thfloat
The temperature in units of frequency (or the same units as w).
 Returns
 n_avgfloat or array
Return the number of average photons in thermal equilibrium for a an oscillator with the given frequency and temperature.

linspace_with
(start, stop, num=50, elems=[])[source]¶ Return an array of numbers sampled over specified interval with additional elements added.
Returns num spaced array with elements from elems inserted if not already included in set.
Returned sample array is not evenly spaced if addtional elements are added.
 Parameters
 startint
The starting value of the sequence.
 stopint
The stoping values of the sequence.
 numint, optional
Number of samples to generate.
 elemslist/ndarray, optional
Requested elements to include in array
 Returns
 samplesndadrray
Original equally spaced sample array with additional elements added.

clebsch
(j1, j2, j3, m1, m2, m3)[source]¶ Calculates the ClebschGordon coefficient for coupling (j1,m1) and (j2,m2) to give (j3,m3).
 Parameters
 j1float
Total angular momentum 1.
 j2float
Total angular momentum 2.
 j3float
Total angular momentum 3.
 m1float
zcomponent of angular momentum 1.
 m2float
zcomponent of angular momentum 2.
 m3float
zcomponent of angular momentum 3.
 Returns
 cg_coefffloat
Requested ClebschGordan coefficient.

convert_unit
(value, orig='meV', to='GHz')[source]¶ Convert an energy from unit orig to unit to.
 Parameters
 valuefloat / array
The energy in the old unit.
 origstring
The name of the original unit (“J”, “eV”, “meV”, “GHz”, “mK”)
 tostring
The name of the new unit (“J”, “eV”, “meV”, “GHz”, “mK”)
 Returns
 value_new_unitfloat / array
The energy in the new unit.
File I/O Functions¶

file_data_read
(filename, sep=None)[source]¶ Retrieves an array of data from the requested file.
 Parameters
 filenamestr
Name of file containing reqested data.
 sepstr
Seperator used to store data.
 Returns
 dataarray_like
Data from selected file.

file_data_store
(filename, data, numtype='complex', numformat='decimal', sep=', ')[source]¶ Stores a matrix of data to a file to be read by an external program.
 Parameters
 filenamestr
Name of data file to be stored, including extension.
 data: array_like
Data to be written to file.
 numtypestr {‘complex, ‘real’}
Type of numerical data.
 numformatstr {‘decimal’,’exp’}
Format for written data.
 sepstr
Singlecharacter field seperator. Usually a tab, space, comma, or semicolon.
Parallelization¶
This function provides functions for parallel execution of loops and function mappings, using the builtin Python module multiprocessing.

parfor
(func, *args, **kwargs)[source]¶ Executes a multivariable function in parallel on the local machine.
Parallel execution of a forloop over function func for multiple input arguments and keyword arguments.
Note
From QuTiP 3.1, we recommend to use
qutip.parallel_map
instead of this function. Parameters
 funcfunction_type
A function to run in parallel on the local machine. The function ‘func’ accepts a series of arguments that are passed to the function as variables. In general, the function can have multiple input variables, and these arguments must be passed in the same order as they are defined in the function definition. In addition, the user can pass multiple keyword arguments to the function.
 The following keyword argument is reserved:
 num_cpusint
Number of CPU’s to use. Default uses maximum number of CPU’s. Performance degrades if num_cpus is larger than the physical CPU count of your machine.
 Returns
 resultlist
A
list
with length equal to number of input parameters containing the output from func.

parallel_map
(task, values, task_args=(), task_kwargs={}, **kwargs)[source]¶ Parallel execution of a mapping of values to the function task. This is functionally equivalent to:
result = [task(value, *task_args, **task_kwargs) for value in values]
 Parameters
 taska Python function
The function that is to be called for each value in
task_vec
. valuesarray / list
The list or array of values for which the
task
function is to be evaluated. task_argslist / dictionary
The optional additional argument to the
task
function. task_kwargslist / dictionary
The optional additional keyword argument to the
task
function. progress_barProgressBar
Progress bar class instance for showing progress.
 Returns
 resultlist
The result list contains the value of
task(value, *task_args, **task_kwargs)
for each value invalues
.

serial_map
(task, values, task_args=(), task_kwargs={}, **kwargs)[source]¶ Serial mapping function with the same call signature as parallel_map, for easy switching between serial and parallel execution. This is functionally equivalent to:
result = [task(value, *task_args, **task_kwargs) for value in values]
This function work as a dropin replacement of
qutip.parallel_map
. Parameters
 taska Python function
The function that is to be called for each value in
task_vec
. valuesarray / list
The list or array of values for which the
task
function is to be evaluated. task_argslist / dictionary
The optional additional argument to the
task
function. task_kwargslist / dictionary
The optional additional keyword argument to the
task
function. progress_barProgressBar
Progress bar class instance for showing progress.
 Returns
 resultlist
The result list contains the value of
task(value, *task_args, **task_kwargs)
for each value invalues
.
Semidefinite Programming¶
This module implements internaluse functions for semidefinite programming.
IPython Notebook Tools¶
This module contains utility functions for using QuTiP with IPython notebooks.

parfor
(task, task_vec, args=None, client=None, view=None, show_scheduling=False, show_progressbar=False)[source]¶ Call the function
tast
for each value intask_vec
using a cluster of IPython engines. The functiontask
should have the signaturetask(value, args)
ortask(value)
ifargs=None
.The
client
andview
are the IPython.parallel client and loadbalanced view that will be used in the parfor execution. If these areNone
, new instances will be created. Parameters
 task: a Python function
The function that is to be called for each value in
task_vec
. task_vec: array / list
The list or array of values for which the
task
function is to be evaluated. args: list / dictionary
The optional additional argument to the
task
function. For example a dictionary with parameter values. client: IPython.parallel.Client
The IPython.parallel Client instance that will be used in the parfor execution.
 view: a IPython.parallel.Client view
The view that is to be used in scheduling the tasks on the IPython cluster. Preferably a loadbalanced view, which is obtained from the IPython.parallel.Client instance client by calling, view = client.load_balanced_view().
 show_scheduling: bool {False, True}, default False
Display a graph showing how the tasks (the evaluation of
task
for for the value intask_vec1
) was scheduled on the IPython engine cluster. show_progressbar: bool {False, True}, default False
Display a HTMLbased progress bar duing the execution of the parfor loop.
 Returns
 resultlist
The result list contains the value of
task(value, args)
for each value intask_vec
, that is, it should be equivalent to[task(v, args) for v in task_vec]
.

parallel_map
(task, values, task_args=None, task_kwargs=None, client=None, view=None, progress_bar=None, show_scheduling=False, **kwargs)[source]¶ Call the function
task
for each value invalues
using a cluster of IPython engines. The functiontask
should have the signaturetask(value, *args, **kwargs)
.The
client
andview
are the IPython.parallel client and loadbalanced view that will be used in the parfor execution. If these areNone
, new instances will be created. Parameters
 task: a Python function
The function that is to be called for each value in
task_vec
. values: array / list
The list or array of values for which the
task
function is to be evaluated. task_args: list / dictionary
The optional additional argument to the
task
function. task_kwargs: list / dictionary
The optional additional keyword argument to the
task
function. client: IPython.parallel.Client
The IPython.parallel Client instance that will be used in the parfor execution.
 view: a IPython.parallel.Client view
The view that is to be used in scheduling the tasks on the IPython cluster. Preferably a loadbalanced view, which is obtained from the IPython.parallel.Client instance client by calling, view = client.load_balanced_view().
 show_scheduling: bool {False, True}, default False
Display a graph showing how the tasks (the evaluation of
task
for for the value intask_vec1
) was scheduled on the IPython engine cluster. show_progressbar: bool {False, True}, default False
Display a HTMLbased progress bar during the execution of the parfor loop.
 Returns
 resultlist
The result list contains the value of
task(value, task_args, task_kwargs)
for each value invalues
.

version_table
(verbose=False)[source]¶ Print an HTMLformatted table with version numbers for QuTiP and its dependencies. Use it in a IPython notebook to show which versions of different packages that were used to run the notebook. This should make it possible to reproduce the environment and the calculation later on.
 Returns
 version_table: string
Return an HTMLformatted string containing version information for QuTiP dependencies.
Miscellaneous¶

about
()[source]¶ About box for QuTiP. Gives version numbers for QuTiP, NumPy, SciPy, Cython, and MatPlotLib.

simdiag
(ops, evals=True)[source]¶ Simultaneous diagonalization of commuting Hermitian matrices.
 Parameters
 opslist/array
list
orarray
of qobjs representing commuting Hermitian operators.
 Returns
 eigstuple
Tuple of arrays representing eigvecs and eigvals of quantum objects corresponding to simultaneous eigenvectors and eigenvalues for each operator.