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

basis
(N, n=0, offset=0)[source]¶ Generates the vector representation of a Fock state.
Parameters: N : int
Number of Fock states in Hilbert space.
n : int
Integer corresponding to desired number state, defaults to 0 if omitted.
offset : int (default 0)
The lowest number state that is included in the finite number state representation of the state.
Returns: state : qobj
Qobj representing the requested number state
n>
.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]]

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: seq : str / 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.
dim : int (default: 2) / list of ints
Space dimension for each particle: int if there are the same, list if they are different.
Returns: bra : qobj
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: N : int
Number of Fock states in Hilbert space.
alpha : float/complex
Eigenvalue of coherent state.
offset : int (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’.
method : string {‘operator’, ‘analytic’}
Method for generating coherent state.
Returns: state : qobj
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: N : int
Number of Fock states in Hilbert space.
alpha : float/complex
Eigenvalue for coherent state.
offset : int (default 0)
The lowest number state that is included in the finite number state representation of the state.
method : string {‘operator’, ‘analytic’}
Method for generating coherent density matrix.
Returns: dm : qobj
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.
excitations : integer
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: dims : list
A list of the dimensions of each subsystem of a composite quantum system.
excitations : integer
The maximum number of excitations that are to be included in the state space.
n : integer
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: dm : Qobj
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: dims : list
A list of the dimensions of each subsystem of a composite quantum system.
excitations : integer
The maximum number of excitations that are to be included in the state space.
state : list of integers
The state in the number basis representation.
Returns: ket : Qobj
A Qobj instance that represent a Fock state in the exicationnumber restricted state space defined by dims and exciations.

fock
(N, n=0, offset=0)[source]¶ Bosonic Fock (number) state.
Same as
qutip.states.basis
.Parameters: N : int
Number of states in the Hilbert space.
n : int
int
for desired number state, defaults to 0 if omitted.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
(N, n=0, offset=0)[source]¶ Density matrix representation of a Fock state
Constructed via outer product of
qutip.states.fock
.Parameters: N : int
Number of Fock states in Hilbert space.
n : int
int
for desired number state, defaults to 0 if omitted.Returns: dm : qobj
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: N : int (default=3)
Number of qubits in state
Returns: G : qobj
Nqubit GHZstate

maximally_mixed_dm
(N)[source]¶ Returns the maximally mixed density matrix for a Hilbert space of dimension N.
Parameters: N : int
Number of basis states in Hilbert space.
Returns: dm : qobj
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: seq : str / 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.
dim : int (default: 2) / list of ints
Space dimension for each particle: int if there are the same, list if they are different.
Returns: ket : qobj
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: Q : qobj
Ket or bra type quantum object.
Returns: dm : qobj
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: N : int
Number of basis vectors in Hilbert space.
m : int
Integer corresponding to the mth discrete phase phi_m=phi0+2*pi*m/N
phi0 : float (default=0)
Reference phase angle.
Returns: state : qobj
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=0)[source]¶ The projection operator that projects state \(m>\) on state \(n>\).
Parameters: N : int
Number of basis states in Hilbert space.
n, m : float
The number states in the projection.
offset : int (default 0)
The lowest number state that is included in the finite number state representation of the projector.
Returns: oper : qobj
Requested projection operator.

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

singlet_state
()[source]¶ Returns the two particle singletstate:
that is identical to the fourth bell state.
Returns: Bell_state : qobj
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: j : float
The spin of the state ().
m : int
Eigenvalue of the spinj Sz operator.
type : string {‘ket’, ‘bra’, ‘dm’}
Type of state to generate.
Returns: state : qobj
Qobj quantum object for spin state

spin_coherent
(j, theta, phi, type='ket')[source]¶ Generates the spin state j, m>, i.e. the eigenstate of the spinj Sz operator with eigenvalue m.
Parameters: j : float
The spin of the state.
theta : float
Angle from z axis.
phi : float
Angle from x axis.
type : string {‘ket’, ‘bra’, ‘dm’}
Type of state to generate.
Returns: state : qobj
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: dims : list or array
The quantum state dimensions array, as it would appear in a Qobj.
state : list
Current state in the iteration. Used internally.
excitations : integer (None)
Restrict state space to states with excitation numbers below or equal to this value.
idx : integer
Current index in the iteration. Used internally.
Returns: state_number : list
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: dims : list or array
The quantum state dimensions array, as it would appear in a Qobj.
state : list
State number array.
Returns: idx : int
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: dims : list or array
The quantum state dimensions array, as it would appear in a Qobj.
index : integer
The index of the state in standard enumeration ordering.
Returns: state : list
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: dims : list or array
The quantum state dimensions array, as it would appear in a Qobj.
state : list
State number array.
Returns: state :
qutip.Qobj.qobj
The state as a
qutip.Qobj.qobj
instance.

thermal_dm
(N, n, method='operator')[source]¶ Density matrix for a thermal state of n particles
Parameters: N : int
Number of basis states in Hilbert space.
n : float
Expectation value for number of particles in thermal state.
method : string {‘operator’, ‘analytic’}
string
that sets the method used to generate the thermal state probabilitiesReturns: dm : qobj
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: Nmax : int
Maximum charge state to consider.
Nmin : int (default = Nmax)
Lowest charge state to consider.
frac : float (default = 1)
Specify fractional charge if needed.
Returns: C : Qobj
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: N : int
Dimension of Hilbert space.
Returns: oper : qobj
Qobj for raising operator.
offset : int (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: N : int
Dimension of Hilbert space.
offset : int (default 0)
The lowest number state that is included in the finite number state representation of the operator.
Returns: oper : qobj
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: N : int
Dimension of Hilbert space.
alpha : float/complex
Displacement amplitude.
offset : int (default 0)
The lowest number state that is included in the finite number state representation of the operator.
Returns: oper : qobj
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: dims : list
A list of the dimensions of each subsystem of a composite quantum system.
excitations : integer
The maximum number of excitations that are to be included in the state space.
Returns: a_ops : list 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: dims : list
A list of the dimensions of each subsystem of a composite quantum system.
excitations : integer
The maximum number of excitations that are to be included in the state space.
state : list of integers
The state in the number basis representation.
Returns: op : Qobj
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: j : float
Spin of operator
args : str
Which operator to return ‘x’,’y’,’z’,’+’,’‘. If no args given, then output is [‘x’,’y’,’z’]
Returns: jmat : qobj / 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: N : int
The dimension of the Hilbert space.
offset : int (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
(N)[source]¶ Identity operator
Parameters: N : int or list of ints
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.Returns: oper : qobj
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.]]

identity
(N)[source]¶ Identity operator. Alternative name to
qeye
.Parameters: N : int or list of ints
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.Returns: oper : qobj
Identity operator Qobj.

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

phase
(N, phi0=0)[source]¶ Singlemode PeggBarnett phase operator.
Parameters: N : int
Number of basis states in Hilbert space.
phi0 : float
Reference phase.
Returns: oper : qobj
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: N : int
Number of Fock states in Hilbert space.
offset : int (default 0)
The lowest number state that is included in the finite number state representation of the operator.
Returns: oper : qobj
Position operator as Qobj.

qdiags
(diagonals, offsets, dims=None, shape=None)[source]¶ Constructs an operator from an array of diagonals.
Parameters: diagonals : sequence of array_like
Array of elements to place along the selected diagonals.
offsets : sequence of ints
 Sequence for diagonals to be set:
 k=0 main diagonal
 k>0 kth upper diagonal
 k<0 kth lower diagonal
dims : list, optional
Dimensions for operator
shape : list, tuple, optional
Shape of operator. If omitted, a square operator large enough to contain the diagonals is generated.
See also
scipy.sparse.diags
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
(N)[source]¶ Zero operator
Parameters: N : int or list of ints
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.Returns: qzero : qobj
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
>>> sigmam() 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: j : float
Spin of operator
Returns: op : Qobj
qobj
representation of the operator.

spin_Jy
(j)[source]¶ Spinj y operator
Parameters: j : float
Spin of operator
Returns: op : Qobj
qobj
representation of the operator.

spin_Jz
(j)[source]¶ Spinj z operator
Parameters: j : float
Spin of operator
Returns: op : Qobj
qobj
representation of the operator.

spin_Jm
(j)[source]¶ Spinj annihilation operator
Parameters: j : float
Spin of operator
Returns: op : Qobj
qobj
representation of the operator.

spin_Jp
(j)[source]¶ Spinj creation operator
Parameters: j : float
Spin of operator
Returns: op : Qobj
qobj
representation of the operator.

squeeze
(N, z, offset=0)[source]¶ Singlemode Squeezing operator.
Parameters: N : int
Dimension of hilbert space.
z : float/complex
Squeezing parameter.
offset : int (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.
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.
z : float/complex
Squeezing parameter.
Returns: oper :
qutip.qobj.Qobj
Squeezing operator.
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)[source]¶ Creates a random NxN density matrix.
Parameters: N : int, ndarray, list
If int, then shape of output operator. If list/ndarray then eigenvalues of generated density matrix.
density : float
Density between [0,1] of output density matrix.
dims : list
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[N]].
Returns: oper : qobj
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)[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: N : int
Dimension of the density operator to be returned.
dims : list
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[N]].
rank : int or None
Rank of the sampled density operator. If None, a fullrank density operator is generated.
Returns: rho : Qobj
An N × N density operator sampled from the Ginibre or HilbertSchmidt distribution.

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

rand_herm
(N, density=0.75, dims=None, pos_def=False)[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: N : int, list/ndarray
If int, then shape of output operator. If list/ndarray then eigenvalues of generated operator.
density : float
Density between [0,1] of output Hermitian operator.
dims : list
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N],[N]].
pos_def : bool (default=False)
Return a positive semidefinite matrix (by diagonal dominance).
Returns: oper : qobj
NxN Hermitian quantum operator.

rand_ket
(N, density=1, dims=None)[source]¶ Creates a random Nx1 sparse ket vector.
Parameters: N : int
Number of rows for output quantum operator.
density : float
Density between [0,1] of output ket state.
dims : list
Leftdimensions of quantum object. Used for specifying tensor structure. Default is dims=[[N]].
Returns: oper : qobj
Nx1 ket state quantum operator.

rand_ket_haar
(N=2, dims=None)[source]¶ Returns a Haar random pure state of dimension
dim
by applying a Haar random unitary to a fixed pure state.Parameters: N : int
Dimension of the state vector to be returned.
dims : list of ints, or None
Leftdimensions of the resultant quantum object. If None, [N] is used.
Returns: psi : Qobj
A random state vector drawn from the Haar measure.

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

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

rand_super
(N=5, dims=None)[source]¶ Returns a randomly drawn superoperator acting on operators acting on N dimensions.
Parameters: N : int
Square root of the dimension of the superoperator to be returned.
dims : list
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)[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: N : int
Square root of the dimension of the superoperator to be returned.
enforce_tp : bool
If True, the tracepreserving condition of [BCSZ08] is enforced; otherwise only complete positivity is enforced.
rank : int or None
Rank of the sampled superoperator. If None, a fullrank superoperator is generated.
dims : list
Dimensions of quantum object. Used for specifying tensor structure. Default is dims=[[[N],[N]], [[N],[N]]].
Returns: rho : Qobj
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 [R1] .
[R1]  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: H : qobj
System Hamiltonian.
c_ops : array_like
A
list
orarray
of collapse operators.Returns: L : qobj
Liouvillian superoperator.

spost
(A)[source]¶ Superoperator formed from postmultiplication by operator A
Parameters: A : qobj
Quantum operator for post multiplication.
Returns: super : qobj
Superoperator formed from input qauntum object.

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

sprepost
(A, B)[source]¶ Superoperator formed from premultiplication by operator A and post multiplication of operator B.
Parameters: A : Qobj
Quantum operator for premultiplication.
B : Qobj
Quantum operator for postmultiplication.
Returns: super : Qobj
Superoperator formed from input quantum objects.

lindblad_dissipator
(a, b=None, data_only=False)[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: a : qobj
Left part of collapse operator.
b : qobj (optional)
Right part of collapse operator. If not specified, b defaults to a.
Returns: D : qobj
Lindblad dissipator superoperator.
Superoperator Representations¶
This module implements transformations between superoperator representations, including supermatrix, Kraus, Choi and Chi (process) matrix formalisms.

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_oper : Qobj
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: choi : Qobj
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_super
(q_oper)[source]¶ Converts a Qobj representing a quantum map to the supermatrix (Liouville) representation.
Parameters: q_oper : Qobj
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: superop : Qobj
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)[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_oper : Qobj
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]
.Returns: kraus_ops : list 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.
Functions acting on states and operators¶
Tensor¶
Module for the creation of composite quantum objects via the tensor product.

tensor
(*args)[source]¶ Calculates the tensor product of input operators.
Parameters: args : array_like
list
orarray
of quantum objects for tensor product.Returns: obj : qobj
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: args : array_like
list
orarray
of quantum objects withtype="super"
.Returns: obj : qobj
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: pairs : tuple
One or more tuples
(i, j)
indicating that thei
andj
dimensions of the original qobj should be contracted.Returns: cqobj : Qobj
The original Qobj with all named index pairs contracted away.
Expectation Values¶

expect
(oper, state)[source]¶ Calculates the expectation value for operator(s) and state(s).
Parameters: oper : qobj/arraylike
A single or a list or operators for expectation value.
state : qobj/arraylike
A single or a list of quantum states or density matrices.
Returns: expt : float/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
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.
mask : list / array
A mask that selects which subsystems should be transposed.
method : str
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).
Returns: rho_pr:
qutip.qobj
A density matrix with the selected subsystems transposed.
Entropy Functions¶

concurrence
(rho)[source]¶ Calculate the concurrence entanglement measure for a twoqubit state.
Parameters: state : qobj
Ket, bra, or density matrix for a twoqubit state.
Returns: concur : float
Concurrence
References
[R2] http://en.wikipedia.org/wiki/Concurrence_(quantum_computing)

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: rho : qobj
Density matrix of composite object
selB : int/list
Selected components for density matrix B
base : {e,2}
Base of logarithm.
sparse : {False,True}
Use sparse eigensolver.
Returns: ent_cond : float
Value of conditional entropy

entropy_linear
(rho)[source]¶ Linear entropy of a density matrix.
Parameters: rho : qobj
sensity matrix or ket/bra vector.
Returns: entropy : float
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: rho : qobj
Density matrix for composite quantum systems
selA : int/list
int or list of first selected density matrix components.
selB : int/list
int or list of second selected density matrix components.
base : {e,2}
Base of logarithm.
sparse : {False,True}
Use sparse eigensolver.
Returns: ent_mut : float
Mutual information between selected components.

entropy_vn
(rho, base=2.718281828459045, sparse=False)[source]¶ VonNeumann entropy of density matrix
Parameters: rho : qobj
Density matrix.
base : {e,2}
Base of logarithm.
sparse : {False,True}
Use sparse eigensolver.
Returns: entropy : float
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: A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns: fid : float
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: A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
tol : float
Tolerance used by sparse eigensolver, if used. (0=Machine precision)
sparse : {False, True}
Use sparse eigensolver.
Returns: tracedist : float
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: A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns: dist : float
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: A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns: angle : float
Bures angle between density matrices.

hilbert_dist
(A, B)[source]¶ Returns the HilbertSchmidt distance between two density matrices A & B.
Parameters: A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
Returns: dist : float
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: A : Qobj
Quantum object representing a superoperator.
target : Qobj
Quantum object representing the target unitary; the inverse is applied before evaluating the fidelity.
Returns: fid : float
Fidelity pseudometric between A and the identity superoperator, or between A and the target superunitary.
Continous 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: basis : list
List of operators that defines the basis for the correlation matrix.
rho : Qobj
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_mat : ndarray
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: basis : list
List of operators that defines the basis for the covariance matrix.
rho : Qobj
Density matrix for which to calculate the covariance matrix.
symmetrized : bool {True, False}
Flag indicating whether the symmetrized (default) or nonsymmetrized correlation matrix is to be calculated.
Returns: corr_mat : ndarray
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: a1 : Qobj
Field operator for mode 1.
a2 : Qobj
Field operator for mode 2.
rho : Qobj
Density matrix for which to calculate the covariance matrix.
Returns: cov_mat : ndarray
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)[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: a1 : Qobj
Field operator for mode 1.
a2 : Qobj
Field operator for mode 2.
rho : Qobj
Density matrix for which to calculate the covariance matrix.
Returns: corr_mat : ndarray
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)[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: a1 : Qobj
Field operator for mode 1.
a2 : Qobj
Field operator for mode 2.
R : ndarray
The quadrature correlation matrix.
rho : Qobj
Density matrix for which to calculate the covariance matrix.
Returns: cov_mat : ndarray
A 2dimensional array of covariance values.

logarithmic_negativity
(V)[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: V : 2d array
The covariance matrix.
Returns: N : float
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, rho0, tlist, e_ops=[], args={}, options=None, progress_bar=<qutip.ui.progressbar.BaseProgressBar object>, _safe_mode=True)[source]¶ Schrodinger equation evolution of a state vector for a given Hamiltonian.
Evolve the state vector or density matrix (rho0) using a given Hamiltonian (H), by integrating the set of ordinary differential equations that define the system.
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.
Parameters: H :
qutip.qobj
system Hamiltonian, or a callback function for timedependent Hamiltonians.
rho0 :
qutip.qobj
initial density matrix or state vector (ket).
tlist : list / array
list of times for \(t\).
e_ops : list of
qutip.qobj
/ callback function singlesingle operator or list of operators for which to evaluate expectation values.
args : dictionary
dictionary of parameters for timedependent Hamiltonians and collapse operators.
options :
qutip.Qdeoptions
with options for the ODE solver.
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 or density matrices 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.
Master Equation¶
This module provides solvers for the Lindblad master equation and von Neumann equation.

mesolve
(H, rho0, tlist, c_ops=[], e_ops=[], args={}, 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).
tlist : list / array
list of times for \(t\).
c_ops : list of
qutip.Qobj
single collapse operator, or list of collapse operators, or a list of Liouvillian superoperators.
e_ops : list of
qutip.Qobj
/ callback function singlesingle operator or list of operators for which to evaluate expectation values.
args : dictionary
dictionary of parameters for timedependent Hamiltonians and collapse operators.
options :
qutip.Options
with options for the solver.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for showing the progress of the simulation.
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.
Monte Carlo Evolution¶

mcsolve
(H, psi0, tlist, c_ops=[], e_ops=[], ntraj=None, args={}, options=None, progress_bar=True, map_func=None, map_kwargs=None, _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
System Hamiltonian.
psi0 :
qutip.Qobj
Initial state vector
tlist : array_like
Times at which results are recorded.
ntraj : int
Number of trajectories to run.
c_ops : array_like
single collapse operator or
list
orarray
of collapse operators.e_ops : array_like
single operator or
list
orarray
of operators for calculating expectation values.args : dict
Arguments for timedependent Hamiltonian and collapse operator terms.
options : Options
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.
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).
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: H : qobj/function_type
System Hamiltonian.
rho0 :
qutip.qobj
Initial state density matrix.
tlist : list/array
list
of times for \(t\).c_op_list : list of
qutip.qobj
list
ofqutip.qobj
collapse operators.e_ops : list of
qutip.qobj
list
ofqutip.qobj
operators for which to evaluate expectation values.Returns: expt_array : array
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: L : qobj
Liouvillian of the system.
rho0 : qobj
Initial state vector or density matrix.
Returns: eseries :
qutip.eseries
eseries
represention of the system dynamics.
BlochRedfield Master Equation¶

brmesolve
(H, psi0, tlist, a_ops, e_ops=[], spectra_cb=[], c_ops=[], args={}, options=<qutip.solver.Options object>, _safe_mode=True)[source]¶ Solve the dynamics for a system using the BlochRedfield master equation.
Note
This solver does not currently support timedependent Hamiltonians.
Parameters: H :
qutip.Qobj
System Hamiltonian.
rho0 / psi0: :class:`qutip.Qobj`
Initial density matrix or state vector (ket).
tlist : list / array
List of times for \(t\).
a_ops : list of
qutip.qobj
List of system operators that couple to bath degrees of freedom.
e_ops : list of
qutip.qobj
/ callback functionList of operators for which to evaluate expectation values.
c_ops : list of
qutip.qobj
List of system collapse operators.
args : dictionary
Placeholder for future implementation, kept for API consistency.
options :
qutip.solver.Options
Options for the solver.
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.

bloch_redfield_tensor
(H, a_ops, spectra_cb, c_ops=[], use_secular=True)[source]¶ Calculate the BlochRedfield tensor for a system given a set of operators and corresponding spectral functions that describes the system’s coupling to its environment.
Note
This tensor generation requires a timeindependent Hamiltonian.
Parameters: H :
qutip.qobj
System Hamiltonian.
a_ops : list of
qutip.qobj
List of system operators that couple to the environment.
spectra_cb : list of callback functions
List of callback functions that evaluate the noise power spectrum at a given frequency.
c_ops : list of
qutip.qobj
List of system collapse operators.
use_secular : bool
Flag (True of False) that indicates if the secular approximation should be used.
Returns: R, kets:
qutip.Qobj
, list ofqutip.Qobj
R is the BlochRedfield tensor and kets is a list eigenstates of the Hamiltonian.

bloch_redfield_solve
(R, ekets, rho0, tlist, e_ops=[], options=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.
ekets : array of
qutip.qobj
Array of kets that make up a basis tranformation for the eigenbasis.
rho0 :
qutip.qobj
Initial density matrix.
tlist : list / array
List of times for \(t\).
e_ops : list of
qutip.qobj
/ callback functionList of operators for which to evaluate expectation values.
options :
qutip.Qdeoptions
Options for the ODE solver.
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.
Floquet States and FloquetMarkov Master Equation¶

fmmesolve
(H, rho0, tlist, c_ops=[], e_ops=[], spectra_cb=[], T=None, args={}, options=<qutip.solver.Options object>, 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).
tlist : list / array
list of times for \(t\).
c_ops : list of
qutip.qobj
list of collapse operators.
e_ops : list of
qutip.qobj
/ callback functionlist of operators for which to evaluate expectation values.
spectra_cb : list callback functions
List of callback functions that compute the noise power spectrum as a function of frequency for the collapse operators in c_ops.
T : float
The period of the timedependence of the hamiltonian. The default value ‘None’ indicates that the ‘tlist’ spans a single period of the driving.
args : dictionary
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_max : int
The truncation of the number of sidebands (default 5).
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.

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
args : dictionary
dictionary with variables required to evaluate H
T : float
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
.Returns: output : list 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_0 : list of
qutip.qobj
(kets)Floquet modes at \(t\)
f_energies : list
Floquet energies.
t : float
The time at which to evaluate the floquet modes.
H :
qutip.qobj
system Hamiltonian, timedependent with period T
args : dictionary
dictionary with variables required to evaluate H
T : float
The period of the timedependence of the hamiltonian.
Returns: output : list 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_0 : list of
qutip.qobj
(kets)Floquet modes at \(t\)
f_energies : list
Floquet energies.
tlist : array
The list of times at which to evaluate the floquet modes.
H :
qutip.qobj
system Hamiltonian, timedependent with period T
T : float
The period of the timedependence of the hamiltonian.
args : dictionary
dictionary with variables required to evaluate H
Returns: output : nested 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_t : nested list of
qutip.qobj
(kets)A lookuptable of Floquet modes at times precalculated by
qutip.floquet.floquet_modes_table
.t : float
The time for which to evaluate the Floquet modes.
T : float
The period of the timedependence of the hamiltonian.
Returns: output : nested 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_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_t : list of
qutip.qobj
(kets)A list of initial Floquet modes (for time \(t=0\)).
f_energies : array
The Floquet energies.
t : float
The time for which to evaluate the Floquet states.
H :
qutip.qobj
System Hamiltonian, timedependent with period T.
T : float
The period of the timedependence of the hamiltonian.
args : dictionary
Dictionary with variables required to evaluate H.
Returns: output : list
A list of Floquet states for the time \(t\).

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_t : list of
qutip.qobj
(kets)A list of initial Floquet modes (for time \(t=0\)).
f_energies : array
The Floquet energies.
f_coeff : array
The coefficients for Floquet decomposition of the initial wavefunction.
t : float
The time for which to evaluate the Floquet states.
H :
qutip.qobj
System Hamiltonian, timedependent with period T.
T : float
The period of the timedependence of the hamiltonian.
args : dictionary
Dictionary with variables required to evaluate H.
Returns: output :
qutip.qobj
The wavefunction for the time \(t\).

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_states : list of
qutip.qobj
(kets)A list of Floquet modes.
f_energies : array
The Floquet energies.
psi :
qutip.qobj
The wavefunction to decompose in the Floquet state basis.
Returns: output : array
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).
tlist : list / array
list of times for \(t\).
e_ops : list of
qutip.qobj
/ callback functionlist 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.
T : float
The period of the timedependence of the hamiltonian.
args : dictionary
Dictionary with variables required to evaluate H.
Tsteps : integer
The number of time steps in one driving period for which to precalculate the Floquet modes. Tsteps should be an even number.
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.
Stochastic Schrödinger Equation and Master Equation¶
This module contains functions for solving stochastic schrodinger and master equations. The API should not be considered stable, and is subject to change when we work more on optimizing this module for performance and features.

smesolve
(H, rho0, times, c_ops=[], sc_ops=[], e_ops=[], _safe_mode=True, **kwargs)[source]¶ Solve stochastic master equation. Dispatch to specific solvers depending on the value of the solver keyword argument.
Parameters: H :
qutip.Qobj
System Hamiltonian.
rho0 :
qutip.Qobj
Initial density matrix or state vector (ket).
times : list / array
List of times for \(t\). Must be uniformly spaced.
c_ops : list of
qutip.Qobj
Deterministic collapse operator which will contribute with a standard Lindblad type of dissipation.
sc_ops : list 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_ops : list of
qutip.Qobj
/ callback function singlesingle operator or list of operators for which to evaluate expectation values.
kwargs : dictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.Returns: output:
qutip.solver.SolverResult
An instance of the class
qutip.solver.SolverResult
.

ssesolve
(H, psi0, times, sc_ops=[], e_ops=[], _safe_mode=True, **kwargs)[source]¶ Solve the stochastic Schrödinger equation. Dispatch to specific solvers depending on the value of the solver keyword argument.
Parameters: H :
qutip.Qobj
System Hamiltonian.
psi0 :
qutip.Qobj
Initial state vector (ket).
times : list / array
List of times for \(t\). Must be uniformly spaced.
sc_ops : list of
qutip.Qobj
List of stochastic collapse operators. Each stochastic collapse operator will give a deterministic and stochastic contribution to the equation of motion according to how the d1 and d2 functions are defined.
e_ops : list of
qutip.Qobj
Single operator or list of operators for which to evaluate expectation values.
kwargs : dictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.Returns: output:
qutip.solver.SolverResult
An instance of the class
qutip.solver.SolverResult
.

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.
times : list / array
List of times for \(t\). Must be uniformly spaced.
c_ops : list of
qutip.Qobj
Deterministic collapse operator which will contribute with a standard Lindblad type of dissipation.
sc_ops : list 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_ops : list of
qutip.Qobj
/ callback function singlesingle operator or list of operators for which to evaluate expectation values.
kwargs : dictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.Returns: output:
qutip.solver.SolverResult
An instance of the class
qutip.solver.SolverResult
.

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).
times : list / array
List of times for \(t\). Must be uniformly spaced.
c_ops : list of
qutip.Qobj
Deterministic collapse operator which will contribute with a standard Lindblad type of dissipation.
e_ops : list of
qutip.Qobj
/ callback function singlesingle operator or list of operators for which to evaluate expectation values.
kwargs : dictionary
Optional keyword arguments. See
qutip.stochastic.StochasticSolverOptions
.Returns: output:
qutip.solver.SolverResult
An instance of the class
qutip.solver.SolverResult
.
Correlation Functions¶

correlation
(H, state0, tlist, taulist, c_ops, a_op, b_op, solver='me', reverse=False, args={}, options=<qutip.solver.Options object>)[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: H : Qobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
state0 : Qobj
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.
tlist : array_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.
taulist : array_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
c_ops : list
list of collapse operators, may be timedependent for solver choice of me or mc.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
reverse : bool
If True, calculate \(\left<A(t)B(t+\tau)\right>\) instead of \(\left<A(t+\tau)B(t)\right>\).
solver : str
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
options : Options
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_mat : array
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>)[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: H : Qobj
system Hamiltonian.
taulist : array_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
c_ops : list
list of collapse operators.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
reverse : bool
If True, calculate \(\lim_{t \to \infty} \left<A(t)B(t+\tau)\right>\) instead of \(\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>\).
solver : str
choice of solver (me for masterequation and es for exponential series).
options : Options
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_vec : array
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>)[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: H : Qobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
state0 : Qobj
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.
taulist : array_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
c_ops : list
list of collapse operators, may be timedependent for solver choice of me or mc.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
reverse : bool {False, True}
If True, calculate \(\left<A(t)B(t+\tau)\right>\) instead of \(\left<A(t+\tau)B(t)\right>\).
solver : str {‘me’, ‘mc’, ‘es’}
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
options : Options
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_vec : ndarray
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>)[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: H : Qobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
state0 : Qobj
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.
tlist : array_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.
taulist : array_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
c_ops : list
list of collapse operators, may be timedependent for solver choice of me or mc.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
reverse : bool {False, True}
If True, calculate \(\left<A(t)B(t+\tau)\right>\) instead of \(\left<A(t+\tau)B(t)\right>\).
solver : str
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
options : Options
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_mat : ndarray
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>)[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: H : Qobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
rho0 : Qobj
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.
taulist : array_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
c_ops : list
list of collapse operators, may be timedependent for solver choice of me or mc.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
c_op : Qobj
operator C.
solver : str
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
options : Options
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_vec : array
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>)[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: H : Qobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
rho0 : Qobj
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.
tlist : array_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.
taulist : array_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
c_ops : list
list of collapse operators, may be timedependent for solver choice of me or mc.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
c_op : Qobj
operator C.
solver : str
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
options : Options
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_mat : array
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>)[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: H : Qobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
rho0 : Qobj
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.
taulist : array_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
c_ops : list
list of collapse operators, may be timedependent for solver choice of me or mc.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
c_op : Qobj
operator C.
d_op : Qobj
operator D.
solver : str
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
options : Options
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_vec : array
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>)[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: H : Qobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
rho0 : Qobj
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.
tlist : array_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.
taulist : array_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
c_ops : list
list of collapse operators, may be timedependent for solver choice of me or mc.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
c_op : Qobj
operator C.
d_op : Qobj
operator D.
solver : str
choice of solver (me for masterequation, mc for Monte Carlo, and es for exponential series).
options : Options
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_mat : array
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.
wlist : array_like
list of frequencies for \(\omega\).
c_ops : list
list of collapse operators.
a_op : Qobj
operator A.
b_op : Qobj
operator B.
solver : str
choice of solver (es for exponential series and pi for psuedoinverse).
use_pinv : bool
For use with the pi solver: if True use numpy’s pinv method, otherwise use a generic solver.
Returns: spectrum : array
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.
wlist : array_like
list of frequencies for \(\omega\).
c_ops : list of
qutip.qobj
list of collapse operators.
a_op :
qutip.qobj
operator A.
b_op :
qutip.qobj
operator B.
use_pinv : bool
If True use numpy’s pinv method, otherwise use a generic solver.
Returns: spectrum : array
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.
wlist : array_like
list of frequencies for \(\omega\).
c_ops : list of
qutip.qobj
list of collapse operators.
a_op :
qutip.qobj
operator A.
b_op :
qutip.qobj
operator B.
use_pinv : bool
If True use numpy’s pinv method, otherwise use a generic solver.
Returns: spectrum : array
An array with spectrum \(S(\omega)\) for the frequencies specified in wlist.

spectrum_correlation_fft
(taulist, y)[source]¶ Calculate the power spectrum corresponding to a twotime correlation function using FFT.
Parameters: tlist : array_like
list/array of times \(t\) which the correlation function is given.
y : array_like
list/array of correlations corresponding to time delays \(t\).
Returns: w, S : tuple
Returns an array of angular frequencies ‘w’ and the corresponding onesided power spectrum ‘S(w)’.

coherence_function_g1
(H, state0, taulist, c_ops, a_op, solver='me', args={}, options=<qutip.solver.Options object>)[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: H : Qobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
state0 : Qobj
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.
taulist : array_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
c_ops : list
list of collapse operators, may be timedependent for solver choice of me or mc.
a_op : Qobj
operator A.
solver : str
choice of solver (me for masterequation and es for exponential series).
options : Options
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, G1 : tuple
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>)[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: H : Qobj
system Hamiltonian, may be timedependent for solver choice of me or mc.
state0 : Qobj
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.
taulist : array_like
list of times for \(\tau\). taulist must be positive and contain the element 0.
c_ops : list
list of collapse operators, may be timedependent for solver choice of me or mc.
a_op : Qobj
operator A.
solver : str
choice of solver (me for masterequation and es for exponential series).
options : Options
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, G2 : tuple
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=[], **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: A : qobj
A Hamiltonian or Liouvillian operator.
c_op_list : list
A list of collapse operators.
method : str {‘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_info : bool, optional, default = False
Return a dictionary of solverspecific infomation about the solution and how it was obtained.
sparse : bool, 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_rcm : bool, optional, default = False
Use reverse CuthillMckee reordering to minimize fillin in the LU factorization of the Liouvillian.
use_wbm : bool, 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.weight : float, 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.
x0 : ndarray, optional
ITERATIVE ONLY. Initial guess for solution vector.
maxiter : int, optional, default=1000
ITERATIVE ONLY. Maximum number of iterations to perform.
tol : float, optional, default=1e12
ITERATIVE ONLY. Tolerance used for terminating solver.
permc_spec : str, 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_precond : bool 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_factor : float, 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_tol : float, 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_thresh : float, 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_MILU : str, 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: dm : qobj
Steady state density matrix.
info : dict, 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: A : qobj
A Hamiltonian or Liouvillian operator.
c_op_list : list
A list of collapse operators.
return_info : bool, optional, default = False
Return a dictionary of solverspecific infomation about the solution and how it was obtained.
use_rcm : bool, optional, default = False
Use reverse CuthillMckee reordering to minimize fillin in the LU factorization of the Liouvillian.
use_wbm : bool, 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.weight : float, 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.
method : str, 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_spec : str, 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_factor : float, 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_tol : float, 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_thresh : float, 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_MILU : str, optional, default = ‘smilu_2’
Selects the incomplete LU decomposition method algoithm used in creating the preconditoner. Should only be used by advanced users.
Returns: lu : object
Returns a SuperLU object representing iLU preconditioner.
info : dict, optional
Dictionary containing solverspecific information.
Propagators¶

propagator
(H, t, c_op_list=[], args={}, options=None, unitary_mode='batch', parallel=False, progress_bar=None, **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: H : qobj 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
).t : float or arraylike
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/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’).
parallel : bool {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: a : qobj
Instance representing the propagator \(U(t)\).
Timedependent problems¶

rhs_generate
(H, c_ops, args={}, options=<qutip.solver.Options object>, 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: H : qobj
System Hamiltonian.
c_ops : list
list
of collapse operators.args : dict
Arguments for timedependent Hamiltonian and collapse operator terms.
options : Options
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.
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: state : qobj
A state vector or density matrix.
xvec : array_like
xcoordinates at which to calculate the Wigner function.
yvec : array_like
ycoordinates at which to calculate the Wigner function.
g : float
Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2).
Returns: Q : array
Values representing the Qfunction calculated over the specified range [xvec,yvec].

spin_q_function
(rho, theta, phi)[source]¶ Husimi Qfunction for spins.
Parameters: state : qobj
A state vector or density matrix for a spinj quantum system.
theta : array_like
thetacoordinates at which to calculate the Q function.
phi : array_like
phicoordinates at which to calculate the Q function.
Returns: Q, THETA, PHI : 2darray
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: state : qobj
A state vector or density matrix for a spinj quantum system.
theta : array_like
thetacoordinates at which to calculate the Q function.
phi : array_like
phicoordinates at which to calculate the Q function.
Returns: W, THETA, PHI : 2darray
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: state : qobj
A state vector or density matrix.
xvec : array_like
xcoordinates at which to calculate the Wigner function.
yvec : array_like
ycoordinates at which to calculate the Wigner function. Does not apply to the ‘fft’ method.
g : float
Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2).
method : string {‘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.
sparse : bool {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.
parfor : bool {False, True}
Flag for calculating the Laguerre polynomial based Wigner function method=’laguerre’ in parallel using the parfor function.
Returns: W : array
Values representing the Wigner function calculated over the specified range [xvec,yvec].
yvex : array
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: rho : qobj
Input density matrix or superoperator.
xlabels : list of strings or False
list of x labels
ylabels : list of strings or False
list of y labels
title : string
title of the plot (optional)
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
cmap : a matplotlib colormap instance
Color map to use when plotting.
label_top : bool
If True, xaxis labels will be placed on top, otherwise they will appear below the plot.
Returns: fig, ax : tuple
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: M : Matrix of Qobj
The matrix to visualize
xlabels : list of strings
list of x labels
ylabels : list of strings
list of y labels
title : string
title of the plot (optional)
limits : list/array with two float numbers
The zaxis limits [min, max] (optional)
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
Returns: fig, ax : tuple
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: M : Matrix of Qobj
The matrix to visualize
xlabels : list of strings
list of x labels
ylabels : list of strings
list of y labels
title : string
title of the plot (optional)
limits : list/array with two float numbers
The zaxis limits [min, max] (optional)
phase_limits : list/array with two float numbers
The phaseaxis (colorbar) limits [min, max] (optional)
ax : a 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, ax : tuple
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_list : List of Qobj
A list of Hamiltonians.
 labels : List of string
A list of labels for each Hamiltonian
 show_ylabels : Bool (default False)
Show y labels to the left of energy levels of the initial Hamiltonian.
 N : int
The number of energy levels to plot
 figsize : tuple (int,int)
The size of the figure (width, height).
 fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
 ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
Returns: fig, ax : tuple
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.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
title : string
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).
Returns: fig, ax : tuple
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.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
axes : a 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).
cmap : a matplotlib cmap instance
The colormap.
alpha_max : float
The span of the x and y coordinates (both [alpha_max, alpha_max]).
colorbar : bool
Whether (True) or not (False) a colorbar should be attached to the Wigner function graph.
method : string {‘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’).
Returns: fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce the figure.

plot_wigner
(rho, fig=None, ax=None, figsize=(8, 4), cmap=None, alpha_max=7.5, colorbar=False, method='iterative', 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.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
ax : a 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).
cmap : a matplotlib cmap instance
The colormap.
alpha_max : float
The span of the x and y coordinates (both [alpha_max, alpha_max]).
colorbar : bool
Whether (True) or not (False) a colorbar should be attached to the Wigner function graph.
method : string {‘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’).
Returns: fig, ax : tuple
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: theta : float
Angle with respect to zaxis
phi : float
Angle in xy plane
values : array
Data set to be plotted
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
save : bool {False , True}
Whether to save the figure or not
Returns: fig, ax : tuple
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: ket : Qobj
Pure state for plotting.
splitting : int
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_iteration : int or pair of ints (default (3,2))
Number of particles to be shown as tick labels, for first (vertical) and last (horizontal) particles, respectively.
fig : a matplotlib figure instance
The figure canvas on which the plot will be drawn.
ax : a 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, ax : tuple
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: ket : Qobj
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_iteration : int (default 1)
Helper lines to be drawn on plot. Show tiles for 2*grid_iteration particles vs all others.
legend_iteration : int (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.
fig : a matplotlib figure instance
The figure canvas on which the plot will be drawn.
ax : a 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, ax : tuple
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.
ylabels : list of strings
The yaxis labels. List should be of the same length as results.
title : string
The title of the figure.
show_legend : bool
Whether or not to show the legend.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
axes : a 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).
Returns: fig, ax : tuple
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: P : matrix
Distribution values as a meshgrid matrix.
THETA : matrix
Meshgrid matrix for the theta coordinate.
PHI : matrix
Meshgrid matrix for the phi coordinate.
fig : a matplotlib figure instance
The figure canvas on which the plot will be drawn.
ax : a 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, ax : tuple
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: P : matrix
Distribution values as a meshgrid matrix.
THETA : matrix
Meshgrid matrix for the theta coordinate.
PHI : matrix
Meshgrid matrix for the phi coordinate.
fig : a matplotlib figure instance
The figure canvas on which the plot will be drawn.
ax : a 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, ax : tuple
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: theta : list/array
Polar angles
phi : list/array
Azimuthal angles
args : list/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: U : Qobj
Transformation operator. Can be calculated using QuTiP propagator function.
op_basis_list : list
A list of Qobj’s representing the basis states.
Returns: chi : array
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: chi : array
Input QPT chi matrix.
lbls_list : list
List of labels for QPT plot axes.
title : string
Plot title.
fig : figure instance
User defined figure instance used for generating QPT plot.
axes : list of figure axis instance
User defined figure axis instance (list of two axes) used for generating QPT plot.
Returns: fig, ax : tuple
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: chi : array
Input QPT chi matrix.
lbls_list : list
List of labels for QPT plot axes.
title : string
Plot title.
fig : figure instance
User defined figure instance used for generating QPT plot.
ax : figure 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, ax : tuple
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: result : qobj
Quantum object for operator describing the rotation.

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

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

sqrtnot
(N=None, target=0)[source]¶ Singlequbit square root NOT gate.
Returns: result : qobj
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_gate : qobj
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: theta : float
Phase rotation angle.
Returns: phase_gate : qobj
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: theta : float
Phase rotation angle.
N : integer
The number of qubits in the target space.
control : integer
The index of the control qubit.
target : integer
The index of the target qubit.
Returns: U : qobj
Quantum object representation of controlled phase gate.

cnot
(N=None, control=0, target=1)[source]¶ Quantum object representing the CNOT gate.
Returns: cnot_gate : qobj
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_gate : qobj
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_gate : qobj
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_gate : qobj
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_gate : qobj
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_gate : qobj
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_gate : qobj
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_gate : qobj
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_gate : qobj
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_gate : qobj
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: result : qobj
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: U : Qobj
Arbitrary singlequbit gate.
N : integer
The number of qubits in the target space.
control : integer
The index of the first control qubit.
target : integer
The index of the target qubit.
control_value : integer (1)
The state of the control qubit that activates the gate U.
Returns: result : qobj
Quantum object representing the controlledU gate.

globalphase
(theta, N=1)[source]¶ Returns quantum object representing the global phase shift gate.
Parameters: theta : float
Phase rotation angle.
Returns: phase_gate : qobj
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: q : qobj
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_list : list
List of gates implementing the quantum circuit.
left_to_right : Boolean
Check if multiplication is to be done from left to right.
Returns: U_overall : qobj
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: U : Qobj
The onequbit gate
N : integer
The number of qubits in the target space.
target : integer
The index of the target qubit.
Returns: gate : qobj
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: U : Qobj
The twoqubit gate
N : integer
The number of qubits in the target space.
control : integer
The index of the control qubit.
target : integer
The index of the target qubit.
targets : list
List of target qubits.
Returns: gate : qobj
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: U : Qobj
The threequbit gate
N : integer
The number of qubits in the target space.
controls : list
The list of the control qubits.
target : integer
The index of the target qubit.
Returns: gate : qobj
Quantum object representation of Nqubit gate.
Qubits¶
Algorithms¶
This module provides the circuit implementation for Quantum Fourier Transform.

qft
(N=1)[source]¶ Quantum Fourier Transform operator on N qubits.
Parameters: N : int
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.
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: dynmaps : list 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).
times : array_like
list of times \(t_n\) at which to compute \(\rho(t_n)\). Must be uniformily spaced.
e_ops : list of
qutip.Qobj
/ callback functionsingle operator or list of operators for which to evaluate expectation values.
learningtimes : array_like
list of times \(t_k\) for which we have knowledge of the dynamical maps \(E(t_k)\).
tensors : array_like
optional list of precomputed tensors \(T_k\)
kwargs : dictionary
Optional keyword arguments. See
qutip.nonmarkov.ttm.TTMSolverOptions
.Returns: output:
qutip.solver.Result
An instance of the class
qutip.solver.Result
.
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, len(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: drift : Qobj or list of Qobj
the underlying dynamics generator of the system can provide list (of length num_tslots) for time dependent drift
ctrls : List of Qobj
a list of control dynamics generators. These are scaled by the amplitudes to alter the overall dynamics
initial : Qobj
starting point for the evolution. Typically the identity matrix
target : Qobj
target transformation, e.g. gate or state, for the time evolution
num_tslots : integer or None
number of timeslots. None implies that timeslots will be given in the tau array
evo_time : float or None
total time for the evolution None implies that timeslots will be given in the tau array
tau : array[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_lbound : float 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_ubound : float 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_targ : float
Fidelity error target. Pulse optimisation will terminate when the fidelity error falls below this value
mim_grad : float
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_iter : integer
Maximum number of iterations of the optimisation algorithm
max_wall_time : float
Maximum allowed elapsed time for the optimisation algorithm
alg : string
Algorithm to use in pulse optimisation. Options are:
‘GRAPE’ (default)  GRadient Ascent Pulse Engineering ‘CRAB’  Chopped RAndom Basis
alg_params : Dictionary
options that are specific to the algorithm see above
optim_params : Dictionary
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_method : string
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_params : dict
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_alg : string
Deprecated. Use optim_method.
max_metric_corr : integer
Deprecated. Use method_params instead
accuracy_factor : float
Deprecated. Use method_params instead
dyn_type : string
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_params : dict
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_type : string
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_params : dict
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_type : string
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_params : dict
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_option : string
Deprecated. Pass in fid_params instead.
fid_err_scale_factor : float
Deprecated. Use scale_factor key in fid_params instead.
tslot_type : string
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_params : dict
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_mode : string
Deprecated. Use tslot_type instead.
init_pulse_type : string
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_params : dict
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_scaling : float
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_offset : float
Linear offset for the pulse. That is this value will be added to any initial / guess pulses generated.
ramping_pulse_type : string
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_params : dict
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_level : integer
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_ext : string 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_stats : boolean
if set to True then statistics for the optimisation run will be generated  accessible through attributes of the stats object
Returns: opt : OptimResult
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, len(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_d : Qobj 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_c : Qobj
a list of control Hamiltonians. These are scaled by the amplitudes to alter the overall dynamics
U_0 : Qobj
starting point for the evolution. Typically the identity matrix
U_targ : Qobj
target transformation, e.g. gate or state, for the time evolution
num_tslots : integer or None
number of timeslots. None implies that timeslots will be given in the tau array
evo_time : float or None
total time for the evolution None implies that timeslots will be given in the tau array
tau : array[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_lbound : float 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_ubound : float 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_targ : float
Fidelity error target. Pulse optimisation will terminate when the fidelity error falls below this value
mim_grad : float
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_iter : integer
Maximum number of iterations of the optimisation algorithm
max_wall_time : float
Maximum allowed elapsed time for the optimisation algorithm
alg : string
Algorithm to use in pulse optimisation. Options are:
‘GRAPE’ (default)  GRadient Ascent Pulse Engineering ‘CRAB’  Chopped RAndom Basis
alg_params : Dictionary
options that are specific to the algorithm see above
optim_params : Dictionary
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_method : string
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_params : dict
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_alg : string
Deprecated. Use optim_method.
max_metric_corr : integer
Deprecated. Use method_params instead
accuracy_factor : float
Deprecated. Use method_params instead
phase_option : string
determines how global phase is treated in fidelity calculations (fid_type=’UNIT’ only). Options:
PSU  global phase ignored SU  global phase included
dyn_params : dict
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_params : dict
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_params : dict
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_type : string
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_params : dict
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_mode : string
Deprecated. Use tslot_type instead.
init_pulse_type : string
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_params : dict
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_scaling : float
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_offset : float
Linear offset for the pulse. That is this value will be added to any initial / guess pulses generated.
ramping_pulse_type : string
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_params : dict
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_level : integer
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_ext : string 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_stats : boolean
if set to True then statistics for the optimisation run will be generated  accessible through attributes of the stats object
Returns: opt : OptimResult
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: drift : Qobj or list of Qobj
the underlying dynamics generator of the system can provide list (of length num_tslots) for time dependent drift
ctrls : List of Qobj
a list of control dynamics generators. These are scaled by the amplitudes to alter the overall dynamics
initial : Qobj
starting point for the evolution. Typically the identity matrix
target : Qobj
target transformation, e.g. gate or state, for the time evolution
num_tslots : integer or None
number of timeslots. None implies that timeslots will be given in the tau array
evo_time : float or None
total time for the evolution None implies that timeslots will be given in the tau array
tau : array[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_lbound : float 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_ubound : float 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_targ : float
Fidelity error target. Pulse optimisation will terminate when the fidelity error falls below this value
mim_grad : float
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_iter : integer
Maximum number of iterations of the optimisation algorithm
max_wall_time : float
Maximum allowed elapsed time for the optimisation algorithm
alg : string
Algorithm to use in pulse optimisation. Options are:
‘GRAPE’ (default)  GRadient Ascent Pulse Engineering ‘CRAB’  Chopped RAndom Basis
alg_params : Dictionary
options that are specific to the algorithm see above
optim_params : Dictionary
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_method : string
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_params : dict
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_alg : string
Deprecated. Use optim_method.
max_metric_corr : integer
Deprecated. Use method_params instead
accuracy_factor : float
Deprecated. Use method_params instead
dyn_type : string
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_params : dict
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_type : string
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_params : dict
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_type : string
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_params : dict
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_option : string
Deprecated. Pass in fid_params instead.
fid_err_scale_factor : float
Deprecated. Use scale_factor key in fid_params instead.
tslot_type : string
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_params : dict
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_mode : string
Deprecated. Use tslot_type instead.
init_pulse_type : string
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_params : dict
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_scaling : float
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_offset : float
Linear offset for the pulse. That is this value will be added to any initial / guess pulses generated.
ramping_pulse_type : string
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_params : dict
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_level : integer
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_stats : boolean
if set to True then statistics for the optimisation run will be generated  accessible through attributes of the stats object
Returns: opt : Optimizer
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, len(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: drift : Qobj or list of Qobj
the underlying dynamics generator of the system can provide list (of length num_tslots) for time dependent drift
ctrls : List of Qobj
a list of control dynamics generators. These are scaled by the amplitudes to alter the overall dynamics
initial : Qobj
starting point for the evolution. Typically the identity matrix
target : Qobj
target transformation, e.g. gate or state, for the time evolution
num_tslots : integer or None
number of timeslots. None implies that timeslots will be given in the tau array
evo_time : float or None
total time for the evolution None implies that timeslots will be given in the tau array
tau : array[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_lbound : float 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_ubound : float 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_targ : float
Fidelity error target. Pulse optimisation will terminate when the fidelity error falls below this value
max_iter : integer
Maximum number of iterations of the optimisation algorithm
max_wall_time : float
Maximum allowed elapsed time for the optimisation algorithm
alg_params : Dictionary
options that are specific to the algorithm see above
optim_params : Dictionary
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_scaling : float
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_coeffs : integer
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_method : string
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_params : dict
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_type : string
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_params : dict
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_type : string
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_params : dict
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_type : string
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_params : dict
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_type : string
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_params : dict
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_type : string
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_params : dict
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_action : string
Determines how the guess pulse is applied to the pulse generated by the basis expansion. Options are: MODULATE, ADD Default is MODULATE
pulse_scaling : float
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_offset : float
Linear offset for the pulse. That is this value will be added to any guess pulses generated.
ramping_pulse_type : string
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_params : dict
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_level : integer
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_ext : string 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_stats : boolean
if set to True then statistics for the optimisation run will be generated  accessible through attributes of the stats object
Returns: opt : OptimResult
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, len(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_d : Qobj 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_c : Qobj
a list of control Hamiltonians. These are scaled by the amplitudes to alter the overall dynamics
U_0 : Qobj
starting point for the evolution. Typically the identity matrix
U_targ : Qobj
target transformation, e.g. gate or state, for the time evolution
num_tslots : integer or None
number of timeslots. None implies that timeslots will be given in the tau array
evo_time : float or None
total time for the evolution None implies that timeslots will be given in the tau array
tau : array[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_lbound : float 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_ubound : float 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_targ : float
Fidelity error target. Pulse optimisation will terminate when the fidelity error falls below this value
max_iter : integer
Maximum number of iterations of the optimisation algorithm
max_wall_time : float
Maximum allowed elapsed time for the optimisation algorithm
alg_params : Dictionary
options that are specific to the algorithm see above
optim_params : Dictionary
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_scaling : float
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_coeffs : integer
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_method : string
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_params : dict
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_option : string
determines how global phase is treated in fidelity calculations (fid_type=’UNIT’ only). Options:
PSU  global phase ignored SU  global phase included
dyn_params : dict
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_params : dict
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_params : dict
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_type : string
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_params : dict
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_type : string
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_params : dict
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_action : string
Determines how the guess pulse is applied to the pulse generated by the basis expansion. Options are: MODULATE, ADD Default is MODULATE
pulse_scaling : float
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_offset : float
Linear offset for the pulse. That is this value will be added to any guess pulses generated.
ramping_pulse_type : string
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_params : dict
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_level : integer
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_ext : string 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_stats : boolean
if set to True then statistics for the optimisation run will be generated  accessible through attributes of the stats object
Returns: opt : OptimResult
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 timeslotThese 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 0These are the periodic options
SINE  Sine wave SQUARE  Square wave SAW  Saw tooth wave TRIANGLE  Triangular waveIf a Dynamics object is passed in then this is used in instantiate the PulseGen, meaning that some timeslot and amplitude properties are copied over.
Utilitiy 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: A : csc_matrix, csr_matrix
Input graph in CSC or CSR matrix format
start : int
Staring node for BFS traversal.
Returns: order : array
Order in which nodes are traversed from starting node.
levels : array
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: A : qobj, csr_matrix, csc_matrix
Input quantum object or csr_matrix.
Returns: degree : array
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: A : csc_matrix, csr_matrix
Input sparse CSC or CSR sparse matrix format.
sym : bool {False, True}
Flag to set whether input matrix is symmetric.
Returns: perm : array
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: A : sparse matrix
Input matrix
perm_type : str {‘row’, ‘column’}
Type of permutation to generate.
Returns: perm : array
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: A : csc_matrix
Input matrix
perm_type : str {‘row’, ‘column’}
Type of permutation to generate.
Returns: perm : array
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: w : float or array
Frequency of the oscillator.
w_th : float
The temperature in units of frequency (or the same units as w).
Returns: n_avg : float 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: start : int
The starting value of the sequence.
stop : int
The stoping values of the sequence.
num : int, optional
Number of samples to generate.
elems : list/ndarray, optional
Requested elements to include in array
Returns: samples : ndadrray
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: j1 : float
Total angular momentum 1.
j2 : float
Total angular momentum 2.
j3 : float
Total angular momentum 3.
m1 : float
zcomponent of angular momentum 1.
m2 : float
zcomponent of angular momentum 2.
m3 : float
zcomponent of angular momentum 3.
Returns: cg_coeff : float
Requested ClebschGordan coefficient.

convert_unit
(value, orig='meV', to='GHz')[source]¶ Convert an energy from unit orig to unit to.
Parameters: value : float / array
The energy in the old unit.
orig : string
The name of the original unit (“J”, “eV”, “meV”, “GHz”, “mK”)
to : string
The name of the new unit (“J”, “eV”, “meV”, “GHz”, “mK”)
Returns: value_new_unit : float / 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: filename : str
Name of file containing reqested data.
sep : str
Seperator used to store data.
Returns: data : array_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: filename : str
Name of data file to be stored, including extension.
data: array_like
Data to be written to file.
numtype : str {‘complex, ‘real’}
Type of numerical data.
numformat : str {‘decimal’,’exp’}
Format for written data.
sep : str
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: func : function_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_cpus : int
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: result : list
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: 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.progress_bar : ProgressBar
Progress bar class instance for showing progress.
Returns: result : list
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: 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.progress_bar : ProgressBar
Progress bar class instance for showing progress.
Returns: result : list
The result list contains the value of
task(value, *task_args, **task_kwargs)
for each value invalues
.
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: result : list
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: result : list
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]¶ Simulateous diagonalization of communting Hermitian matrices..
Parameters: ops : list/array
list
orarray
of qobjs representing commuting Hermitian operators.Returns: eigs : tuple
Tuple of arrays representing eigvecs and eigvals of quantum objects corresponding to simultaneous eigenvectors and eigenvalues for each operator.