Classes
Qobj
 class Qobj(arg=None, dims=None, type=None, copy=True, superrep=None, isherm=None, isunitary=None)[source]
A class for representing quantum objects, such as quantum operators and states.
The Qobj class is the QuTiP representation of quantum operators and state vectors. This class also implements math operations +,,* between Qobj instances (and / by a Cnumber), as well as a collection of common operator/state operations. The Qobj constructor optionally takes a dimension
list
and/or shapelist
as arguments. Parameters:
 inpt: array_like
Data for vector/matrix representation of the quantum object.
 dims: list
Dimensions of object used for tensor products.
 type: {‘bra’, ‘ket’, ‘oper’, ‘operatorket’, ‘operatorbra’, ‘super’}
The type of quantum object to be represented.
 shape: list
Shape of underlying data structure (matrix shape).
 copy: bool
Flag specifying whether Qobj should get a copy of the input data, or use the original.
 Attributes:
 dataarray_like
Sparse matrix characterizing the quantum object.
 dimslist
List of dimensions keeping track of the tensor structure.
 shapelist
Shape of the underlying data array.
 typestr
Type of quantum object: ‘bra’, ‘ket’, ‘oper’, ‘operatorket’, ‘operatorbra’, or ‘super’.
 superrepstr
Representation used if type is ‘super’. One of ‘super’ (Liouville form) or ‘choi’ (Choi matrix with tr = dimension).
 ishermbool
Indicates if quantum object represents Hermitian operator.
 isunitarybool
Indictaes if quantum object represents unitary operator.
 iscpbool
Indicates if the quantum object represents a map, and if that map is completely positive (CP).
 ishpbool
Indicates if the quantum object represents a map, and if that map is hermicity preserving (HP).
 istpbool
Indicates if the quantum object represents a map, and if that map is trace preserving (TP).
 iscptpbool
Indicates if the quantum object represents a map that is completely positive and trace preserving (CPTP).
 isketbool
Indicates if the quantum object represents a ket.
 isbrabool
Indicates if the quantum object represents a bra.
 isoperbool
Indicates if the quantum object represents an operator.
 issuperbool
Indicates if the quantum object represents a superoperator.
 isoperketbool
Indicates if the quantum object represents an operator in column vector form.
 isoperbrabool
Indicates if the quantum object represents an operator in row vector form.
Methods
copy()
Create copy of Qobj
conj()
Conjugate of quantum object.
cosm()
Cosine of quantum object.
dag()
Adjoint (dagger) of quantum object.
dnorm()
Diamond norm of quantum operator.
dual_chan()
Dual channel of quantum object representing a CP map.
eigenenergies(sparse=False, sort=’low’, eigvals=0, tol=0, maxiter=100000)
Returns eigenenergies (eigenvalues) of a quantum object.
eigenstates(sparse=False, sort=’low’, eigvals=0, tol=0, maxiter=100000)
Returns eigenenergies and eigenstates of quantum object.
expm()
Matrix exponential of quantum object.
full(order=’C’)
Returns dense array of quantum object data attribute.
groundstate(sparse=False, tol=0, maxiter=100000)
Returns eigenvalue and eigenket for the groundstate of a quantum object.
inv()
Return a Qobj corresponding to the matrix inverse of the operator.
matrix_element(bra, ket)
Returns the matrix element of operator between bra and ket vectors.
norm(norm=’tr’, sparse=False, tol=0, maxiter=100000)
Returns norm of a ket or an operator.
permute(order)
Returns composite qobj with indices reordered.
proj()
Computes the projector for a ket or bra vector.
ptrace(sel)
Returns quantum object for selected dimensions after performing partial trace.
sinm()
Sine of quantum object.
sqrtm()
Matrix square root of quantum object.
tidyup(atol=1e12)
Removes small elements from quantum object.
tr()
Trace of quantum object.
trans()
Transpose of quantum object.
transform(inpt, inverse=False)
Performs a basis transformation defined by inpt matrix.
trunc_neg(method=’clip’)
Removes negative eigenvalues and returns a new Qobj that is a valid density operator.
unit(norm=’tr’, sparse=False, tol=0, maxiter=100000)
Returns normalized quantum object.
 check_herm()[source]
Check if the quantum object is hermitian.
 Returns:
 ishermbool
Returns the new value of isherm property.
 contract(inplace=False)[source]
Contract subspaces of the tensor structure which are 1D. Not defined on superoperators. If all dimensions are scalar, a Qobj of dimension [[1], [1]] is returned, i.e. _multiple_ scalar dimensions are contracted, but one is left.
 Parameters:
 inplace: bool, optional
If
True
, modify the dimensions in place. IfFalse
, return a copied object.
 Returns:
 out:
Qobj
Quantum object with dimensions contracted. Will be
self
ifinplace
isTrue
.
 out:
 cosm()[source]
Cosine of a quantum operator.
Operator must be square.
 Returns:
 oper
qutip.Qobj
Matrix cosine of operator.
 oper
 Raises:
 TypeError
Quantum object is not square.
Notes
Uses the Q.expm() method.
 data_as(format=None, copy=True)[source]
Matrix from quantum object.
 Parameters:
 formatstr, default: None
Type of the output, “ndarray” for
Dense
, “csr_matrix” forCSR
. A ValueError will be raised if the format is not supported. copybool {False, True}
Whether to return a copy
 Returns:
 datanumpy.ndarray, scipy.sparse.matrix_csr, etc.
Matrix in the type of the underlying libraries.
 diag()[source]
Diagonal elements of quantum object.
 Returns:
 diagsarray
Returns array of
real
values if operators is Hermitian, otherwisecomplex
values are returned.
 dnorm(B=None)[source]
Calculates the diamond norm, or the diamond distance to another operator.
 Parameters:
 B
qutip.Qobj
or None If B is not None, the diamond distance d(A, B) = dnorm(A  B) between this operator and B is returned instead of the diamond norm.
 B
 Returns:
 dfloat
Either the diamond norm of this operator, or the diamond distance from this operator to B.
 eigenenergies(sparse=False, sort='low', eigvals=0, tol=0, maxiter=100000)[source]
Eigenenergies of a quantum object.
Eigenenergies (eigenvalues) are defined for operators or superoperators only.
 Parameters:
 sparsebool
Use sparse Eigensolver
 sortstr
Sort eigenvalues ‘low’ to high, or ‘high’ to low.
 eigvalsint
Number of requested eigenvalues. Default is all eigenvalues.
 tolfloat
Tolerance used by sparse Eigensolver (0=machine precision). The sparse solver may not converge if the tolerance is set too low.
 maxiterint
Maximum number of iterations performed by sparse solver (if used).
 Returns:
 eigvalsarray
Array of eigenvalues for operator.
Notes
The sparse eigensolver is much slower than the dense version. Use sparse only if memory requirements demand it.
 eigenstates(sparse=False, sort='low', eigvals=0, tol=0, maxiter=100000, phase_fix=None)[source]
Eigenstates and eigenenergies.
Eigenstates and eigenenergies are defined for operators and superoperators only.
 Parameters:
 sparsebool
Use sparse Eigensolver
 sortstr
Sort eigenvalues (and vectors) ‘low’ to high, or ‘high’ to low.
 eigvalsint
Number of requested eigenvalues. Default is all eigenvalues.
 tolfloat
Tolerance used by sparse Eigensolver (0 = machine precision). The sparse solver may not converge if the tolerance is set too low.
 maxiterint
Maximum number of iterations performed by sparse solver (if used).
 phase_fixint, None
If not None, set the phase of each kets so that ket[phase_fix,0] is real positive.
 Returns:
 eigvalsarray
Array of eigenvalues for operator.
 eigvecsarray
Array of quantum operators representing the oprator eigenkets. Order of eigenkets is determined by order of eigenvalues.
Notes
The sparse eigensolver is much slower than the dense version. Use sparse only if memory requirements demand it.
 expm(dtype=<class 'qutip.core.data.dense.Dense'>)[source]
Matrix exponential of quantum operator.
Input operator must be square.
 Parameters:
 dtypetype
The datalayer type that should be output. As the matrix exponential is almost dense, this defaults to outputting dense matrices.
 Returns:
 oper
qutip.Qobj
Exponentiated quantum operator.
 oper
 Raises:
 TypeError
Quantum operator is not square.
 full(order='C', squeeze=False)[source]
Dense array from quantum object.
 Parameters:
 orderstr {‘C’, ‘F’}
Return array in C (default) or Fortran ordering.
 squeezebool {False, True}
Squeeze output array.
 Returns:
 dataarray
Array of complex data from quantum objects data attribute.
 groundstate(sparse=False, tol=0, maxiter=100000, safe=True)[source]
Ground state Eigenvalue and Eigenvector.
Defined for quantum operators or superoperators only.
 Parameters:
 sparsebool
Use sparse Eigensolver
 tolfloat
Tolerance used by sparse Eigensolver (0 = machine precision). The sparse solver may not converge if the tolerance is set too low.
 maxiterint
Maximum number of iterations performed by sparse solver (if used).
 safebool (default=True)
Check for degenerate ground state
 Returns:
 eigvalfloat
Eigenvalue for the ground state of quantum operator.
 eigvec
qutip.Qobj
Eigenket for the ground state of quantum operator.
Notes
The sparse eigensolver is much slower than the dense version. Use sparse only if memory requirements demand it.
 inv(sparse=False)[source]
Matrix inverse of a quantum operator
Operator must be square.
 Returns:
 oper
qutip.Qobj
Matrix inverse of operator.
 oper
 Raises:
 TypeError
Quantum object is not square.
 logm()[source]
Matrix logarithm of quantum operator.
Input operator must be square.
 Returns:
 oper
qutip.Qobj
Logarithm of the quantum operator.
 oper
 Raises:
 TypeError
Quantum operator is not square.
 matrix_element(bra, ket)[source]
Calculates a matrix element.
Gives the matrix element for the quantum object sandwiched between a bra and ket vector.
 Parameters:
 bra
qutip.Qobj
Quantum object of type ‘bra’ or ‘ket’
 ket
qutip.Qobj
Quantum object of type ‘ket’.
 bra
 Returns:
 elemcomplex
Complex valued matrix element.
Notes
It is slightly more computationally efficient to use a ket vector for the ‘bra’ input.
 norm(norm=None, kwargs=None)[source]
Norm of a quantum object.
Default norm is L2norm for kets and tracenorm for operators. Other ket and operator norms may be specified using the norm parameter.
 Parameters:
 normstr
Which type of norm to use. Allowed values for vectors are ‘l2’ and ‘max’. Allowed values for matrices are ‘tr’ for the trace norm, ‘fro’ for the Frobenius norm, ‘one’ and ‘max’.
 kwargsdict
Additional keyword arguments to pass on to the relevant norm solver. See details for each norm function in
data.norm
.
 Returns:
 normfloat
The requested norm of the operator or state quantum object.
 overlap(other)[source]
Overlap between two state vectors or two operators.
Gives the overlap (inner product) between the current bra or ket Qobj and and another bra or ket Qobj. It gives the HilbertSchmidt overlap when one of the Qobj is an operator/density matrix.
 Parameters:
 other
qutip.Qobj
Quantum object for a state vector of type ‘ket’, ‘bra’ or density matrix.
 other
 Returns:
 overlapcomplex
Complex valued overlap.
 Raises:
 TypeError
Can only calculate overlap between a bra, ket and density matrix quantum objects.
 permute(order)[source]
Permute the tensor structure of a quantum object. For example,
qutip.tensor(x, y).permute([1, 0])
will give the same result asqutip.tensor(y, x)
andqutip.tensor(a, b, c).permute([1, 2, 0])
will be the same asqutip.tensor(b, c, a)
For regular objects (bras, kets and operators) we expect
order
to be a flat list of integers, which specifies the new order of the tensor product.For superoperators, we expect
order
to be something like[[0, 2], [1, 3]]
which tells us to permute according to [0, 2, 1, 3], and then group indices according to the length of each sublist. As another example, permuting a superoperator with dimensions of[[[1, 2, 3], [1, 2, 3]], [[1, 2, 3], [1, 2, 3]]]
by anorder
[[0, 3], [1, 4], [2, 5]]
should give a new object with dimensions[[[1, 1], [2, 2], [3, 3]], [[1, 1], [2, 2], [3, 3]]]
. Parameters:
 orderlist
List of indices specifying the new tensor order.
 Returns:
 P
qutip.Qobj
Permuted quantum object.
 P
 proj()[source]
Form the projector from a given ket or bra vector.
 Parameters:
 Q
qutip.Qobj
Input bra or ket vector
 Q
 Returns:
 P
qutip.Qobj
Projection operator.
 P
 ptrace(sel, dtype=None)[source]
Take the partial trace of the quantum object leaving the selected subspaces. In other words, trace out all subspaces which are _not_ passed.
This is typically a function which acts on operators; bras and kets will be promoted to density matrices before the operation takes place since the partial trace is inherently undefined on pure states.
For operators which are currently being represented as states in the superoperator formalism (i.e. the object has type operatorket or operatorbra), the partial trace is applied as if the operator were in the conventional form. This means that for any operator x,
operator_to_vector(x).ptrace(0) == operator_to_vector(x.ptrace(0))
and similar for operatorbra.The story is different for full superoperators. In the formalism that QuTiP uses, if an operator has dimensions (dims) of [[2, 3], [2, 3]] then it can be represented as a state on a Hilbert space of dimensions [2, 3, 2, 3], and a superoperator would be an operator which acts on this joint space. This function performs the partial trace on superoperators by letting the selected components refer to elements of the _joint_ _space_, and then returns a regular operator (of type oper).
 Parameters:
 selint or iterable of int
An
int
orlist
of components to keep after partial trace. The selected subspaces will _not_ be reordered, no matter order they are supplied to ptrace.
 Returns:
 oper
qutip.Qobj
Quantum object representing partial trace with selected components remaining.
 oper
 purity()[source]
Calculate purity of a quantum object.
 Returns:
 state_purityfloat
Returns the purity of a quantum object. For a pure state, the purity is 1. For a mixed state of dimension d, 1/d<=purity<1.
 sinm()[source]
Sine of a quantum operator.
Operator must be square.
 Returns:
 oper
qutip.Qobj
Matrix sine of operator.
 oper
 Raises:
 TypeError
Quantum object is not square.
Notes
Uses the Q.expm() method.
 sqrtm(sparse=False, tol=0, maxiter=100000)[source]
Sqrt of a quantum operator. Operator must be square.
 Parameters:
 sparsebool
Use sparse eigenvalue/vector solver.
 tolfloat
Tolerance used by sparse solver (0 = machine precision).
 maxiterint
Maximum number of iterations used by sparse solver.
 Returns:
 oper
qutip.Qobj
Matrix square root of operator.
 oper
 Raises:
 TypeError
Quantum object is not square.
Notes
The sparse eigensolver is much slower than the dense version. Use sparse only if memory requirements demand it.
 tidyup(atol=None)[source]
Removes small elements from the quantum object.
 Parameters:
 atolfloat
Absolute tolerance used by tidyup. Default is set via qutip global settings parameters.
 Returns:
 oper
qutip.Qobj
Quantum object with small elements removed.
 oper
 to(data_type)[source]
Convert the underlying data store of this Qobj into a different storage representation.
The different storage representations available are the “datalayer types” which are known to qutip.data.to. By default, these are qutip.data.Dense and qutip.data.CSR, which respectively construct a dense matrix store and a compressed sparse row one. Certain algorithms and operations may be faster or more accurate when using a more appropriate data store.
If the data store is already in the format requested, the function returns self. Otherwise, it returns a copy of itself with the data store in the new type.
 Parameters:
 data_typetype
The datalayer type that the data of this Qobj should be converted to.
 Returns:
 Qobj
A new Qobj if a type conversion took place with the data stored in the requested format, or self if not.
 tr()[source]
Trace of a quantum object.
 Returns:
 tracefloat
Returns the trace of the quantum object.
 trans()[source]
Get the matrix transpose of the quantum operator.
 Returns:
 oper
Qobj
Transpose of input operator.
 oper
 transform(inpt, inverse=False)[source]
Basis transform defined by input array.
Input array can be a
matrix
defining the transformation, or alist
of kets that defines the new basis. Parameters:
 inptarray_like
A
matrix
orlist
of kets defining the transformation. inversebool
Whether to return inverse transformation.
 Returns:
 oper
qutip.Qobj
Operator in new basis.
 oper
Notes
This function is still in development.
 trunc_neg(method='clip')[source]
Truncates negative eigenvalues and renormalizes.
Returns a new Qobj by removing the negative eigenvalues of this instance, then renormalizing to obtain a valid density operator.
 Parameters:
 methodstr
Algorithm to use to remove negative eigenvalues. “clip” simply discards negative eigenvalues, then renormalizes. “sgs” uses the SGS algorithm (doi:10/bb76) to find the positive operator that is nearest in the Shatten 2norm.
 Returns:
 oper
qutip.Qobj
A valid density operator.
 oper
 unit(inplace=False, norm=None, kwargs=None)[source]
Operator or state normalized to unity. Uses norm from Qobj.norm().
 Parameters:
 inplacebool
Do an inplace normalization
 normstr
Requested norm for states / operators.
 kwargsdict
Additional keyword arguments to be passed on to the relevant norm function (see
norm
for more details).
 Returns:
 obj
qutip.Qobj
Normalized quantum object. Will be the self object if in place.
 obj
QobjEvo
 class QobjEvo
A class for representing timedependent quantum objects, such as quantum operators and states.
Importantly,
QobjEvo
instances are used to represent such timedependent quantum objects when working with QuTiP solvers.A
QobjEvo
instance may be constructed from one of the following:a callable
f(t: double, args: dict) > Qobj
that returns the value of the quantum object at timet
.a
[Qobj, Coefficient]
pair, whereCoefficient
may also be any item that can be used to construct a coefficient (e.g. a function, a numpy array of coefficient values, a string expression).a
Qobj
(which creates a constantQobjEvo
term).a list of such callables, pairs or
Qobj
s.a
QobjEvo
(in which case a copy is created, all other arguments are ignored exceptargs
which, if passed, replaces the existing arguments).
 Parameters:
 Q_objectcallable, list or Qobj
A specification of the timedepedent quantum object. See the paragraph above for a full description and the examples section below for examples.
 argsdict, optional
A dictionary that contains the arguments for the coefficients. Arguments may be omitted if no function or string coefficients that require arguments are present.
 tlistarraylike, optional
A list of times corresponding to the values of the coefficients supplied as numpy arrays. If no coefficients are supplied as numpy arrays,
tlist
may be omitted, otherwise it is required.The times in
tlist
do not need to be equidistant, but must be sorted.By default, a cubic spline interpolation will be used to interpolate the value of the (numpy array) coefficients at time
t
. If the coefficients are to be treated as step functions, pass the argumentorder=0
(see below). orderint, default=3
Order of the spline interpolation that is to be used to interpolate the value of the (numpy array) coefficients at time
t
.0
use previous or left value. copybool, default=True
Whether to make a copy of the
Qobj
instances supplied in theQ_object
parameter. compressbool, default=True
Whether to compress the
QobjEvo
instance terms after the instance has been created.This sums the constant terms in a single term and combines
[Qobj, coefficient]
pairs with the sameQobj
into a single pair containing the sum of the coefficients.See
compress
. function_style{None, “pythonic”, “dict”, “auto”}
The style of function signature used by callables in
Q_object
. If style isNone
, the value ofqutip.settings.core["function_coefficient_style"]
is used. Otherwise the supplied value overrides the global setting. boundary_conditions2Tuple, str or None, optional
Boundary conditions for spline evaluation. Default value is None. Correspond to bc_type of scipy.interpolate.make_interp_spline. Refer to Scipy’s documentation for further details: https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.make_interp_spline.html
Examples
A
QobjEvo
constructed from a function:def f(t, args): return qutip.qeye(N) * np.exp(args['w'] * t) QobjEvo(f, args={'w': 1j})
For list based
QobjEvo
, the list must consist of :obj`~Qobj` or[Qobj, Coefficient]
pairs:QobjEvo([H0, [H1, coeff1], [H2, coeff2]], args=args)
The coefficients may be specified either using a
Coefficient
object or by a function, string, numpy array or any object that can be passed to thecoefficient
function. See the documentation ofcoefficient
for a full description.An example of a coefficient specified by a function:
def f1_t(t, args): return np.exp(1j * t * args["w1"]) QobjEvo([[H1, f1_t]], args={"w1": 1.})
And of coefficients specified by string expressions:
H = QobjEvo( [H0, [H1, 'exp(1j*w1*t)'], [H2, 'cos(w2*t)']], args={"w1": 1., "w2": 2.} )
Coefficients maybe also be expressed as numpy arrays giving a list of the coefficient values:
tlist = np.logspace(5, 0, 100) H = QobjEvo( [H0, [H1, np.exp(1j * tlist)], [H2, np.cos(2. * tlist)]], tlist=tlist )
The coeffients array must have the same len as the tlist.
A
QobjEvo
may also be built using simple arithmetic operations combiningQobj
withCoefficient
, for example:coeff = qutip.coefficient("exp(1j*w1*t)", args={"w1": 1}) qevo = H0 + H1 * coeff
 Attributes:
 dimslist
List of dimensions keeping track of the tensor structure.
 shape(int, int)
List of dimensions keeping track of the tensor structure.
 typestr
Type of quantum object: ‘bra’, ‘ket’, ‘oper’, ‘operatorket’, ‘operatorbra’, or ‘super’.
 superrepstr
Representation used if type is ‘super’. One of ‘super’ (Liouville form) or ‘choi’ (Choi matrix with tr = dimension).
 arguments()
Update the arguments.
 Parameters:
 _argsdict [optional]
New arguments as a dict. Update args with
arguments(new_args)
. **kwargs
New arguments as a keywors. Update args with
arguments(**new_args)
. .. note::
If both the positional
_args
and keywords are passed new values from both will be used. If a key is present with both, the_args
dict value will take priority.
 compress()
Look for redundance in the
QobjEvo
components:Constant parts, (
Qobj
withoutCoefficient
) will be summed. Pairs[Qobj, Coefficient]
with the sameQobj
are merged.Example:
[[sigmax(), f1], [sigmax(), f2]] > [[sigmax(), f1+f2]]
The
QobjEvo
is transformed inplace. Returns:
 None
 conj()
Get the elementwise conjugation of the quantum object.
 copy()
Return a copy of this QobjEvo
 dag()
Get the Hermitian adjoint of the quantum object.
 expect()
Expectation value of this operator at time
t
with the state. Parameters:
 tfloat
Time of the operator to apply.
 stateQobj
right matrix of the product
 check_realbool (True)
Whether to convert the result to a real when the imaginary part is smaller than the real part by a dactor of
settings.core['rtol']
.
 Returns:
 expectfloat or complex
state.adjoint() @ self @ state
ifstate
is a ket.trace(self @ matrix)
isstate
is an operator or operatorket.
 expect_data()
Expectation is defined as
state.adjoint() @ self @ state
ifstate
is a vector, orstate
is an operator andself
is a superoperator. Ifstate
is an operator andself
is an operator, then expectation istrace(self @ matrix)
.
 isconstant
Does the system change depending on
t
 isoper
Indicates if the system represents an operator.
 issuper
Indicates if the system represents a superoperator.
 linear_map()
Apply mapping to each Qobj contribution.
Example:
QobjEvo([sigmax(), coeff]).linear_map(spre)
gives the same result hasQobjEvo([spre(sigmax()), coeff])
 Returns:
QobjEvo
Modified object
Notes
Does not modify the coefficients, thus
linear_map(conj)
would not give the the conjugate of the QobjEvo. It’s only valid for linear transformations.
 matmul()
Product of this operator at time
t
to the state.self(t) @ state
 Parameters:
 tfloat
Time of the operator to apply.
 stateQobj
right matrix of the product
 Returns:
 productQobj
The result product as a Qobj
 matmul_data()
Compute
out += self(t) @ state
 num_elements
Number of parts composing the system
 tidyup()
Removes small elements from quantum object.
 to()
Convert the underlying data store of all component into the desired storage representation.
The different storage representations available are the “datalayer types”. By default, these are qutip.data.Dense and qutip.data.CSR, which respectively construct a dense matrix store and a compressed sparse row one.
The QobjEvo is transformed inplace.
 Parameters:
 data_typetype
The datalayer type that the data of this Qobj should be converted to.
 Returns:
 None
 to_list()
Restore the QobjEvo to a list form.
 Returns:
 list_qevo: list
The QobjEvo as a list, element are either
Qobj
for constant parts,[Qobj, Coefficient]
for coefficient based term. The original format of theCoefficient
is not restored. Lastly if the original QobjEvo is constructed with a function returning a Qobj, the term is returned as a pair of the original function and args (dict
).
 trans()
Transpose of the quantum object
Bloch sphere
 class Bloch(fig=None, axes=None, view=None, figsize=None, background=False)[source]
Class for plotting data on the Bloch sphere. Valid data can be either points, vectors, or Qobj objects.
 Attributes:
 axesmatplotlib.axes.Axes
User supplied Matplotlib axes for Bloch sphere animation.
 figmatplotlib.figure.Figure
User supplied Matplotlib Figure instance for plotting Bloch sphere.
 font_colorstr, default ‘black’
Color of font used for Bloch sphere labels.
 font_sizeint, default 20
Size of font used for Bloch sphere labels.
 frame_alphafloat, default 0.1
Sets transparency of Bloch sphere frame.
 frame_colorstr, default ‘gray’
Color of sphere wireframe.
 frame_widthint, default 1
Width of wireframe.
 point_colorlist, default [“b”, “r”, “g”, “#CC6600”]
List of colors for Bloch sphere point markers to cycle through, i.e. by default, points 0 and 4 will both be blue (‘b’).
 point_markerlist, default [“o”, “s”, “d”, “^”]
List of point marker shapes to cycle through.
 point_sizelist, default [25, 32, 35, 45]
List of point marker sizes. Note, not all point markers look the same size when plotted!
 sphere_alphafloat, default 0.2
Transparency of Bloch sphere itself.
 sphere_colorstr, default ‘#FFDDDD’
Color of Bloch sphere.
 figsizelist, default [7, 7]
Figure size of Bloch sphere plot. Best to have both numbers the same; otherwise you will have a Bloch sphere that looks like a football.
 vector_colorlist, [“g”, “#CC6600”, “b”, “r”]
List of vector colors to cycle through.
 vector_widthint, default 5
Width of displayed vectors.
 vector_stylestr, default ‘>’
Vector arrowhead style (from matplotlib’s arrow style).
 vector_mutationint, default 20
Width of vectors arrowhead.
 viewlist, default [60, 30]
Azimuthal and Elevation viewing angles.
 xlabellist, default [“$x$”, “”]
List of strings corresponding to +x and x axes labels, respectively.
 xlposlist, default [1.1, 1.1]
Positions of +x and x labels respectively.
 ylabellist, default [“$y$”, “”]
List of strings corresponding to +y and y axes labels, respectively.
 ylposlist, default [1.2, 1.2]
Positions of +y and y labels respectively.
 zlabellist, default [‘$\left0\right>$’, ‘$\left1\right>$’]
List of strings corresponding to +z and z axes labels, respectively.
 zlposlist, default [1.2, 1.2]
Positions of +z and z labels respectively.
 add_annotation(state_or_vector, text, **kwargs)[source]
Add a text or LaTeX annotation to Bloch sphere, parametrized by a qubit state or a vector.
 Parameters:
 state_or_vectorQobj/array/list/tuple
Position for the annotaion. Qobj of a qubit or a vector of 3 elements.
 textstr
Annotation text. You can use LaTeX, but remember to use raw string e.g. r”$langle x rangle$” or escape backslashes e.g. “$\langle x \rangle$”.
 kwargs
Options as for mplot3d.axes3d.text, including: fontsize, color, horizontalalignment, verticalalignment.
 add_arc(start, end, fmt='b', steps=None, **kwargs)[source]
Adds an arc between two points on a sphere. The arc is set to be blue solid curve by default.
The start and end points must be on the same sphere (i.e. have the same radius) but need not be on the unit sphere.
 Parameters:
 startQobj or arraylike
Array with cartesian coordinates of the first point, or a state vector or density matrix that can be mapped to a point on or within the Bloch sphere.
 endQobj or arraylike
Array with cartesian coordinates of the second point, or a state vector or density matrix that can be mapped to a point on or within the Bloch sphere.
 fmtstr, default: “b”
A matplotlib format string for rendering the arc.
 stepsint, default: None
The number of segments to use when rendering the arc. The default uses 100 steps times the distance between the start and end points, with a minimum of 2 steps.
 **kwargsdict
Additional parameters to pass to the matplotlib .plot function when rendering this arc.
 add_line(start, end, fmt='k', **kwargs)[source]
Adds a line segment connecting two points on the bloch sphere.
The line segment is set to be a black solid line by default.
 Parameters:
 startQobj or arraylike
Array with cartesian coordinates of the first point, or a state vector or density matrix that can be mapped to a point on or within the Bloch sphere.
 endQobj or arraylike
Array with cartesian coordinates of the second point, or a state vector or density matrix that can be mapped to a point on or within the Bloch sphere.
 fmtstr, default: “k”
A matplotlib format string for rendering the line.
 **kwargsdict
Additional parameters to pass to the matplotlib .plot function when rendering this line.
 add_points(points, meth='s', colors=None, alpha=1.0)[source]
Add a list of data points to bloch sphere.
 Parameters:
 pointsarray_like
Collection of data points.
 meth{‘s’, ‘m’, ‘l’}
Type of points to plot, use ‘m’ for multicolored, ‘l’ for points connected with a line.
 colorsarray_like
Optional array with colors for the points. A single color for meth ‘s’, and list of colors for meth ‘m’
 alphafloat, default=1.
Transparency value for the vectors. Values between 0 and 1.
 .. note::
When using
meth=l
in QuTiP 4.6, the line transparency defaulted to0.75
and there was no way to alter it. When thealpha
parameter was added in QuTiP 4.7, the default becamealpha=1.0
for values ofmeth
.
 add_states(state, kind='vector', colors=None, alpha=1.0)[source]
Add a state vector Qobj to Bloch sphere.
 Parameters:
 stateQobj
Input state vector.
 kind{‘vector’, ‘point’}
Type of object to plot.
 colorsarray_like
Optional array with colors for the states.
 alphafloat, default=1.
Transparency value for the vectors. Values between 0 and 1.
 add_vectors(vectors, colors=None, alpha=1.0)[source]
Add a list of vectors to Bloch sphere.
 Parameters:
 vectorsarray_like
Array with vectors of unit length or smaller.
 colorsarray_like
Optional array with colors for the vectors.
 alphafloat, default=1.
Transparency value for the vectors. Values between 0 and 1.
 save(name=None, format='png', dirc=None, dpin=None)[source]
Saves Bloch sphere to file of type
format
in directorydirc
. Parameters:
 namestr
Name of saved image. Must include path and format as well. i.e. ‘/Users/Paul/Desktop/bloch.png’ This overrides the ‘format’ and ‘dirc’ arguments.
 formatstr
Format of output image.
 dircstr
Directory for output images. Defaults to current working directory.
 dpinint
Resolution in dots per inch.
 Returns:
 File containing plot of Bloch sphere.
 set_label_convention(convention)[source]
Set x, y and z labels according to one of conventions.
 Parameters:
 conventionstring
One of the following:
“original”
“xyz”
“sx sy sz”
“01”
“polarization jones”
“polarization jones letters” see also: https://en.wikipedia.org/wiki/Jones_calculus
“polarization stokes” see also: https://en.wikipedia.org/wiki/Stokes_parameters
 show()[source]
Display Bloch sphere and corresponding data sets.
Notes
When using inline plotting in Jupyter notebooks, any figure created in a notebook cell is displayed after the cell executes. Thus if you create a figure yourself and use it create a Bloch sphere with
b = Bloch(..., fig=fig)
and then callb.show()
in the same cell, then the figure will be displayed twice. If you do create your own figure, the simplest solution to this is to not call.show()
in the cell you create the figure in.
 vector_mutation
Sets the width of the vectors arrowhead
 vector_style
Style of Bloch vectors, default = ‘>’ (or ‘simple’)
 vector_width
Width of Bloch vectors, default = 5
 class Bloch3d(fig=None)[source]
Class for plotting data on a 3D Bloch sphere using mayavi. Valid data can be either points, vectors, or qobj objects corresponding to state vectors or density matrices. for a twostate system (or subsystem).
Notes
The use of mayavi for 3D rendering of the Bloch sphere comes with a few limitations: I) You can not embed a Bloch3d figure into a matplotlib window. II) The use of LaTex is not supported by the mayavi rendering engine. Therefore all labels must be defined using standard text. Of course you can postprocess the generated figures later to add LaTeX using other software if needed.
 Attributes:
 figinstance {None}
User supplied Matplotlib Figure instance for plotting Bloch sphere.
 font_colorstr {‘black’}
Color of font used for Bloch sphere labels.
 font_scalefloat {0.08}
Scale for font used for Bloch sphere labels.
 framebool {True}
Draw frame for Bloch sphere
 frame_alphafloat {0.05}
Sets transparency of Bloch sphere frame.
 frame_colorstr {‘gray’}
Color of sphere wireframe.
 frame_numint {8}
Number of frame elements to draw.
 frame_radiusfloats {0.005}
Width of wireframe.
 point_colorlist {[‘r’, ‘g’, ‘b’, ‘y’]}
List of colors for Bloch sphere point markers to cycle through. i.e. By default, points 0 and 4 will both be blue (‘r’).
 point_modestring {‘sphere’,’cone’,’cube’,’cylinder’,’point’}
Point marker shapes.
 point_sizefloat {0.075}
Size of points on Bloch sphere.
 sphere_alphafloat {0.1}
Transparency of Bloch sphere itself.
 sphere_colorstr {‘#808080’}
Color of Bloch sphere.
 sizelist {[500,500]}
Size of Bloch sphere plot in pixels. Best to have both numbers the same otherwise you will have a Bloch sphere that looks like a football.
 vector_colorlist {[‘r’, ‘g’, ‘b’, ‘y’]}
List of vector colors to cycle through.
 vector_widthint {3}
Width of displayed vectors.
 viewlist {[45,65]}
Azimuthal and Elevation viewing angles.
 xlabellist {
['x>', '']
} List of strings corresponding to +x and x axes labels, respectively.
 xlposlist {[1.07,1.07]}
Positions of +x and x labels respectively.
 ylabellist {
['y>', '']
} List of strings corresponding to +y and y axes labels, respectively.
 ylposlist {[1.07,1.07]}
Positions of +y and y labels respectively.
 zlabellist {
['0>', '1>']
} List of strings corresponding to +z and z axes labels, respectively.
 zlposlist {[1.07,1.07]}
Positions of +z and z labels respectively.
 add_points(points, meth='s', alpha=1.0)[source]
Add a list of data points to bloch sphere.
 Parameters:
 pointsarray/list
Collection of data points.
 methstr {‘s’,’m’}
Type of points to plot, use ‘m’ for multicolored.
 alphafloat, default=1.
Transparency value for the vectors. Values between 0 and 1.
 add_states(state, kind='vector', alpha=1.0)[source]
Add a state vector Qobj to Bloch sphere.
 Parameters:
 stateqobj
Input state vector.
 kindstr {‘vector’,’point’}
Type of object to plot.
 alphafloat, default=1.
Transparency value for the vectors. Values between 0 and 1.
 add_vectors(vectors, alpha=1.0)[source]
Add a list of vectors to Bloch sphere.
 Parameters:
 vectorsarray/list
Array with vectors of unit length or smaller.
 alphafloat, default=1.
Transparency value for the vectors. Values between 0 and 1.
 save(name=None, format='png', dirc=None)[source]
Saves Bloch sphere to file of type
format
in directorydirc
. Parameters:
 namestr
Name of saved image. Must include path and format as well. i.e. ‘/Users/Paul/Desktop/bloch.png’ This overrides the ‘format’ and ‘dirc’ arguments.
 formatstr
Format of output image. Default is ‘png’.
 dircstr
Directory for output images. Defaults to current working directory.
 Returns:
 File containing plot of Bloch sphere.
Distributions
 class QFunc(xvec, yvec, g: float = 1.4142135623730951, memory: float = 1024)[source]
Classbased method of calculating the HusimiQ function of many different quantum states at fixed phasespace points
0.5*g* (xvec + i*yvec)
. This class has slightly higher firstusage costs thanqfunc
, but subsequent operations will be several times faster. However, it can require quite a lot of memory. Call the created object as a function to retrieve the HusimiQ function. Parameters:
 xvec, yvecarray_like
x and ycoordinates at which to calculate the HusimiQ function.
 gfloat, default sqrt(2)
Scaling factor for
a = 0.5 * g * (x + iy)
. The value of g is related to the value of hbar in the commutation relation \([x,\,y] = i\hbar\) via \(\hbar=2/g^2\), so the default corresponds to \(\hbar=1\). memoryreal, default 1024
Size in MB that may be used internally as workspace. This class will raise
MemoryError
if subsequently passed a state of sufficiently large dimension that this bound would be exceeded. In those cases, useqfunc
withprecompute_memory=None
instead to force using the slower, more memoryefficient algorithm.
See also
qfunc
a single function version, which will involve computing several quantities multiple times in order to use less memory.
Examples
Initialise the class for a square set of coordinates, with some states we want to investigate.
>>> xvec = np.linspace(2, 2, 101) >>> states = [qutip.rand_dm(10) for _ in [None]*10] >>> qfunc = qutip.QFunc(xvec, xvec)
Now we can calculate the HusimiQ function over each of the states more efficiently with:
>>> husimiq = np.array([qfunc(state) for state in states])
Solvers
 class SESolver(H, *, options=None)[source]
Bases:
Solver
Schrodinger equation evolution of a state vector or unitary matrix for a given Hamiltonian.
 Parameters:
 H
Qobj
,QobjEvo
System Hamiltonian as a Qobj or QobjEvo for timedependent Hamiltonians. List of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. optionsdict, optional
Options for the solver, see
SESolver.options
and Integrator for a list of all options.
 H
 Attributes:
 stats: dict
Diverse diagnostic statistics of the evolution.
 classmethod add_integrator(integrator, key)
Register an integrator.
 Parameters:
 integratorIntegrator
The ODE solver to register.
 keyslist of str
Values of the method options that refer to this integrator.
 property options
Solver’s options:
 store_final_state: bool, default=False
Whether or not to store the final state of the evolution in the result class.
 store_states: bool, default=None
Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 normalize_output: bool, default=True
Normalize output state to hide ODE numerical errors.
 progress_bar: str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, {}
How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs: dict, default={“chunk_size”: 10}
Arguments to pass to the progress_bar. Qutip’s bars use
chunk_size
. method: str, default=”adams”
Which ordinary differential equation integration method to use.
 run(state0, tlist, *, args=None, e_ops=None)
Do the evolution of the Quantum system.
For a
state0
at timetlist[0]
do the evolution as directed byrhs
and for each time intlist
store the state and/or expectation values in aResult
. The evolution method and stored results are determined byoptions
. Parameters:
 state0
Qobj
Initial state of the evolution.
 tlistlist of double
Time for which to save the results (state and/or expect) of the evolution. The first element of the list is the initial time of the evolution. Each times of the list must be increasing, but does not need to be uniformy distributed.
 argsdict, optional {None}
Change the
args
of the rhs for the evolution. e_opslist {None}
List of Qobj, QobjEvo or callable to compute the expectation values. Function[s] must have the signature f(t : float, state : Qobj) > expect.
 state0
 start(state0, t0)
Set the initial state and time for a step evolution.
options
for the evolutions are read at this step. Parameters:
 state0
Qobj
Initial state of the evolution.
 t0double
Initial time of the evolution.
 state0
 step(t, *, args=None, copy=True)
Evolve the state to
t
and return the state as aQobj
. Parameters:
 tdouble
Time to evolve to, must be higher than the last call.
 argsdict, optional {None}
Update the
args
of the system. The change is effective from the beginning of the interval. Changingargs
can slow the evolution. copybool, optional {True}
Whether to return a copy of the data or the data in the ODE solver.
 .. note::
The state must be initialized first by calling
start
orrun
. Ifrun
is called,step
will continue from the last time and state obtained.
 property sys_dims
Dimensions of the space that the system use:
qutip.basis(sovler.dims)
will create a state with proper dimensions for this solver.
 class MESolver(H, c_ops=None, *, options=None)[source]
Bases:
SESolver
Master equation evolution of a density matrix for a given Hamiltonian and set of collapse operators, or a Liouvillian.
Evolve the density matrix (rho0) using a given Hamiltonian or Liouvillian (H) and an optional set of collapse operators (c_ops), by integrating the set of ordinary differential equations that define the system.
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 the solution of master equations that are not in standard Lindblad form.
 Parameters:
 H
Qobj
,QobjEvo
Possibly timedependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo. List of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. c_opslist of
Qobj
,QobjEvo
Single collapse operator, or list of collapse operators, or a list of Liouvillian superoperators. None is equivalent to an empty list.
 optionsdict, optional
Options for the solver, see
MESolver.options
and Integrator for a list of all options.
 H
 Attributes:
 stats: dict
Diverse diagnostic statistics of the evolution.
 classmethod add_integrator(integrator, key)
Register an integrator.
 Parameters:
 integratorIntegrator
The ODE solver to register.
 keyslist of str
Values of the method options that refer to this integrator.
 property options
Solver’s options:
 store_final_state: bool, default=False
Whether or not to store the final state of the evolution in the result class.
 store_states: bool, default=None
Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 normalize_output: bool, default=True
Normalize output state to hide ODE numerical errors.
 progress_bar: str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, {}
How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs: dict, default={“chunk_size”: 10}
Arguments to pass to the progress_bar. Qutip’s bars use
chunk_size
. method: str, default=”adams”
Which ordinary differential equation integration method to use.
 run(state0, tlist, *, args=None, e_ops=None)
Do the evolution of the Quantum system.
For a
state0
at timetlist[0]
do the evolution as directed byrhs
and for each time intlist
store the state and/or expectation values in aResult
. The evolution method and stored results are determined byoptions
. Parameters:
 state0
Qobj
Initial state of the evolution.
 tlistlist of double
Time for which to save the results (state and/or expect) of the evolution. The first element of the list is the initial time of the evolution. Each times of the list must be increasing, but does not need to be uniformy distributed.
 argsdict, optional {None}
Change the
args
of the rhs for the evolution. e_opslist {None}
List of Qobj, QobjEvo or callable to compute the expectation values. Function[s] must have the signature f(t : float, state : Qobj) > expect.
 state0
 start(state0, t0)
Set the initial state and time for a step evolution.
options
for the evolutions are read at this step. Parameters:
 state0
Qobj
Initial state of the evolution.
 t0double
Initial time of the evolution.
 state0
 step(t, *, args=None, copy=True)
Evolve the state to
t
and return the state as aQobj
. Parameters:
 tdouble
Time to evolve to, must be higher than the last call.
 argsdict, optional {None}
Update the
args
of the system. The change is effective from the beginning of the interval. Changingargs
can slow the evolution. copybool, optional {True}
Whether to return a copy of the data or the data in the ODE solver.
 .. note::
The state must be initialized first by calling
start
orrun
. Ifrun
is called,step
will continue from the last time and state obtained.
 property sys_dims
Dimensions of the space that the system use:
qutip.basis(sovler.dims)
will create a state with proper dimensions for this solver.
 class BRSolver(H, a_ops, c_ops=None, sec_cutoff=0.1, *, options=None)[source]
Bases:
Solver
Bloch Redfield equation evolution of a density matrix for a given Hamiltonian and set of bath coupling operators.
 Parameters:
 H
Qobj
,QobjEvo
Possibly timedependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo. list of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. a_opslist of (a_op, spectra)
Nested list of system operators that couple to the environment, and the corresponding bath spectra.
 a_op
qutip.Qobj
,qutip.QobjEvo
The operator coupling to the environment. Must be hermitian.
 spectra
Coefficient
The corresponding bath spectra. As a Coefficient using an ‘w’ args. Can depend on
t
only if a_op is aqutip.QobjEvo
.SpectraCoefficient
can be used to conver a coefficient depending ont
to one depending onw
.
Example:
a_ops = [ (a+a.dag(), coefficient('w>0', args={'w':0})), (QobjEvo([b+b.dag(), lambda t: ...]), coefficient(lambda t, w: ...), args={"w": 0}), (c+c.dag(), SpectraCoefficient(coefficient(array, tlist=ws))), ]
 a_op
 c_opslist of
Qobj
,QobjEvo
Single collapse operator, or list of collapse operators, or a list of Lindblad dissipator. None is equivalent to an empty list.
 optionsdict, optional
Options for the solver, see
BRSolver.options
and Integrator for a list of all options. sec_cutofffloat {0.1}
Cutoff for secular approximation. Use
1
if secular approximation is not used when evaluating bathcoupling terms.
 H
 Attributes:
 stats: dict
Diverse diagnostic statistics of the evolution.
 classmethod add_integrator(integrator, key)
Register an integrator.
 Parameters:
 integratorIntegrator
The ODE solver to register.
 keyslist of str
Values of the method options that refer to this integrator.
 property options
Options for bloch redfield solver:
 store_final_state: bool, default=False
Whether or not to store the final state of the evolution in the result class.
 store_states: bool, default=None
Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 normalize_output: bool, default=False
Normalize output state to hide ODE numerical errors.
 progress_bar: str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, default=”text”
How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs: dict, default={“chunk_size”:10}
Arguments to pass to the progress_bar. Qutip’s bars use
chunk_size
. tensor_type: str [‘sparse’, ‘dense’, ‘data’], default=”sparse”
Which data type to use when computing the brtensor. With a cutoff ‘sparse’ is usually the most efficient.
 sparse_eigensolver: bool, default=False
Whether to use the sparse eigensolver
 method: str, default=”adams”
Which ODE integrator methods are supported.
 run(state0, tlist, *, args=None, e_ops=None)
Do the evolution of the Quantum system.
For a
state0
at timetlist[0]
do the evolution as directed byrhs
and for each time intlist
store the state and/or expectation values in aResult
. The evolution method and stored results are determined byoptions
. Parameters:
 state0
Qobj
Initial state of the evolution.
 tlistlist of double
Time for which to save the results (state and/or expect) of the evolution. The first element of the list is the initial time of the evolution. Each times of the list must be increasing, but does not need to be uniformy distributed.
 argsdict, optional {None}
Change the
args
of the rhs for the evolution. e_opslist {None}
List of Qobj, QobjEvo or callable to compute the expectation values. Function[s] must have the signature f(t : float, state : Qobj) > expect.
 state0
 start(state0, t0)
Set the initial state and time for a step evolution.
options
for the evolutions are read at this step. Parameters:
 state0
Qobj
Initial state of the evolution.
 t0double
Initial time of the evolution.
 state0
 step(t, *, args=None, copy=True)
Evolve the state to
t
and return the state as aQobj
. Parameters:
 tdouble
Time to evolve to, must be higher than the last call.
 argsdict, optional {None}
Update the
args
of the system. The change is effective from the beginning of the interval. Changingargs
can slow the evolution. copybool, optional {True}
Whether to return a copy of the data or the data in the ODE solver.
 .. note::
The state must be initialized first by calling
start
orrun
. Ifrun
is called,step
will continue from the last time and state obtained.
 property sys_dims
Dimensions of the space that the system use:
qutip.basis(sovler.dims)
will create a state with proper dimensions for this solver.
 class SMESolver(H, sc_ops, heterodyne, *, c_ops=(), options=None)[source]
Bases:
StochasticSolver
Stochastic Master Equation Solver.
 Parameters:
 H
Qobj
,QobjEvo
,QobjEvo
compatible format. System Hamiltonian as a Qobj or QobjEvo for timedependent Hamiltonians. List of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. sc_opslist of (
QobjEvo
,QobjEvo
compatible format) List of stochastic collapse operators.
 heterodynebool, [False]
Whether to use heterodyne or homodyne detection.
 optionsdict, [optional]
Options for the solver, see
SMESolver.options
and SIntegrator for a list of all options.
 H
 classmethod add_integrator(integrator, key)
Register an integrator.
 Parameters:
 integratorIntegrator
The ODE solver to register.
 keyslist of str
Values of the method options that refer to this integrator.
 property options
Options for stochastic solver:
 store_final_state: bool, default=False
Whether or not to store the final state of the evolution in the result class.
 store_states: bool, default=None
Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 store_measurement: bool, [False]
Whether to store the measurement for each trajectories. Storing measurements will also store the wiener process, or brownian noise for each trajectories.
 progress_bar: str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, default=”text”
How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs: dict, default={“chunk_size”:10}
Arguments to pass to the progress_bar. Qutip’s bars use
chunk_size
. keep_runs_results: bool
Whether to store results from all trajectories or just store the averages.
 method: str, default=”rouchon”
Which ODE integrator methods are supported.
 map: str {“serial”, “parallel”, “loky”}, default=”serial”
How to run the trajectories. “parallel” uses concurent module to run in parallel while “loky” use the module of the same name to do so.
 job_timeout: None, int, default=None
Maximum time to compute one trajectory.
 num_cpus: None, int, default=None
Number of cpus to use when running in parallel.
None
detect the number of available cpus. bitgenerator: {None, “MT19937”, “PCG64DXSM”, …}, default=None
Which of numpy.random’s bitgenerator to use. With
None
, your numpy version’s default is used.
 resultclass
alias of
StochasticResult
 run(state, tlist, ntraj=1, *, args=None, e_ops=(), timeout=None, target_tol=None, seed=None)
Do the evolution of the Quantum system.
For a
state
at timetlist[0]
do the evolution as directed byrhs
and for each time intlist
store the state and/or expectation values in aResult
. The evolution method and stored results are determined byoptions
. Parameters:
 state
Qobj
Initial state of the evolution.
 tlistlist of double
Time for which to save the results (state and/or expect) of the evolution. The first element of the list is the initial time of the evolution. Time in the list must be in increasing order, but does not need to be uniformly distributed.
 ntrajint
Number of trajectories to add.
 argsdict, optional {None}
Change the
args
of the rhs for the evolution. e_opslist
list of Qobj or QobjEvo to compute the expectation values. Alternatively, function[s] with the signature f(t, state) > expect can be used.
 timeoutfloat, optional [1e8]
Maximum time in seconds for the trajectories to run. Once this time is reached, the simulation will end even if the number of trajectories is less than
ntraj
. The map function, set in options, can interupt the running trajectory or wait for it to finish. Set to an arbitrary high number to disable. target_tol{float, tuple, list}, optional [None]
Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by
ntraj
. The error is computed using jackknife resampling.target_tol
can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of (atol, rtol) for each e_ops. seed{int, SeedSequence, list} optional
Seed or list of seeds for each trajectories.
 state
 Returns:
 results
qutip.solver.MultiTrajResult
Results of the evolution. States and/or expect will be saved. You can control the saved data in the options.
 results
 start(state, t0, seed=None)
Set the initial state and time for a step evolution.
 Parameters:
 state
Qobj
Initial state of the evolution.
 t0double
Initial time of the evolution.
 seedint, SeedSequence, list, {None}
Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seed, one for each trajectory.
 ..note ::
When using step evolution, only one trajectory can be computed at once.
 state
 step(t, *, args=None, copy=True)
Evolve the state to
t
and return the state as aQobj
. Parameters:
 tdouble
Time to evolve to, must be higher than the last call.
 argsdict, optional {None}
Update the
args
of the system. The change is effective from the beginning of the interval. Changingargs
can slow the evolution. copybool, optional {True}
Whether to return a copy of the data or the data in the ODE solver.
 property sys_dims
Dimensions of the space that the system use:
qutip.basis(sovler.dims)
will create a state with proper dimensions for this solver.
 class SSESolver(H, sc_ops, heterodyne, *, c_ops=(), options=None)[source]
Bases:
StochasticSolver
Stochastic Schrodinger Equation Solver.
 Parameters:
 H
Qobj
,QobjEvo
,QobjEvo
compatible format. System Hamiltonian as a Qobj or QobjEvo for timedependent Hamiltonians. List of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. c_opslist of (
QobjEvo
,QobjEvo
compatible format) Deterministic collapse operator which will contribute with a standard Lindblad type of dissipation.
 sc_opslist of (
QobjEvo
,QobjEvo
compatible format) List of stochastic collapse operators.
 heterodynebool, [False]
Whether to use heterodyne or homodyne detection.
 optionsdict, [optional]
Options for the solver, see
SSESolver.options
and SIntegrator for a list of all options.
 H
 classmethod add_integrator(integrator, key)
Register an integrator.
 Parameters:
 integratorIntegrator
The ODE solver to register.
 keyslist of str
Values of the method options that refer to this integrator.
 property options
Options for stochastic solver:
 store_final_state: bool, default=False
Whether or not to store the final state of the evolution in the result class.
 store_states: bool, default=None
Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 store_measurement: bool, [False]
Whether to store the measurement for each trajectories. Storing measurements will also store the wiener process, or brownian noise for each trajectories.
 progress_bar: str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, default=”text”
How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs: dict, default={“chunk_size”:10}
Arguments to pass to the progress_bar. Qutip’s bars use
chunk_size
. keep_runs_results: bool
Whether to store results from all trajectories or just store the averages.
 method: str, default=”rouchon”
Which ODE integrator methods are supported.
 map: str {“serial”, “parallel”, “loky”}, default=”serial”
How to run the trajectories. “parallel” uses concurent module to run in parallel while “loky” use the module of the same name to do so.
 job_timeout: None, int, default=None
Maximum time to compute one trajectory.
 num_cpus: None, int, default=None
Number of cpus to use when running in parallel.
None
detect the number of available cpus. bitgenerator: {None, “MT19937”, “PCG64DXSM”, …}, default=None
Which of numpy.random’s bitgenerator to use. With
None
, your numpy version’s default is used.
 resultclass
alias of
StochasticResult
 run(state, tlist, ntraj=1, *, args=None, e_ops=(), timeout=None, target_tol=None, seed=None)
Do the evolution of the Quantum system.
For a
state
at timetlist[0]
do the evolution as directed byrhs
and for each time intlist
store the state and/or expectation values in aResult
. The evolution method and stored results are determined byoptions
. Parameters:
 state
Qobj
Initial state of the evolution.
 tlistlist of double
Time for which to save the results (state and/or expect) of the evolution. The first element of the list is the initial time of the evolution. Time in the list must be in increasing order, but does not need to be uniformly distributed.
 ntrajint
Number of trajectories to add.
 argsdict, optional {None}
Change the
args
of the rhs for the evolution. e_opslist
list of Qobj or QobjEvo to compute the expectation values. Alternatively, function[s] with the signature f(t, state) > expect can be used.
 timeoutfloat, optional [1e8]
Maximum time in seconds for the trajectories to run. Once this time is reached, the simulation will end even if the number of trajectories is less than
ntraj
. The map function, set in options, can interupt the running trajectory or wait for it to finish. Set to an arbitrary high number to disable. target_tol{float, tuple, list}, optional [None]
Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by
ntraj
. The error is computed using jackknife resampling.target_tol
can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of (atol, rtol) for each e_ops. seed{int, SeedSequence, list} optional
Seed or list of seeds for each trajectories.
 state
 Returns:
 results
qutip.solver.MultiTrajResult
Results of the evolution. States and/or expect will be saved. You can control the saved data in the options.
 results
 start(state, t0, seed=None)
Set the initial state and time for a step evolution.
 Parameters:
 state
Qobj
Initial state of the evolution.
 t0double
Initial time of the evolution.
 seedint, SeedSequence, list, {None}
Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seed, one for each trajectory.
 ..note ::
When using step evolution, only one trajectory can be computed at once.
 state
 step(t, *, args=None, copy=True)
Evolve the state to
t
and return the state as aQobj
. Parameters:
 tdouble
Time to evolve to, must be higher than the last call.
 argsdict, optional {None}
Update the
args
of the system. The change is effective from the beginning of the interval. Changingargs
can slow the evolution. copybool, optional {True}
Whether to return a copy of the data or the data in the ODE solver.
 property sys_dims
Dimensions of the space that the system use:
qutip.basis(sovler.dims)
will create a state with proper dimensions for this solver.
Monte Carlo Solvers
 class MCSolver(H, c_ops, *, options=None)[source]
Bases:
MultiTrajSolver
Monte Carlo Solver of a state vector \(\psi \rangle\) for a given Hamiltonian and sets of collapse operators. Options for the underlying ODE solver are given by the Options class.
 Parameters:
 H
qutip.Qobj
,qutip.QobjEvo
,list
, callable. System Hamiltonian as a Qobj, QobjEvo. It can also be any input type that QobjEvo accepts (see
qutip.QobjEvo
’s documentation).H
can also be a superoperator (liouvillian) if some collapse operators are to be treated deterministically. c_opslist
A
list
of collapse operators in any input type that QobjEvo accepts (seequtip.QobjEvo
’s documentation). They must be operators even ifH
is a superoperator. optionsdict, [optional]
Options for the evolution.
 H
 classmethod add_integrator(integrator, key)
Register an integrator.
 Parameters:
 integratorIntegrator
The ODE solver to register.
 keyslist of str
Values of the method options that refer to this integrator.
 mc_integrator_class
alias of
MCIntegrator
 property options
Options for monte carlo solver:
 store_final_state: bool, default=False
Whether or not to store the final state of the evolution in the result class.
 store_states: bool, default=None
Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 progress_bar: str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, default=”text”
How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs: dict, default={“chunk_size”:10}
Arguments to pass to the progress_bar. Qutip’s bars use
chunk_size
. keep_runs_results: bool
Whether to store results from all trajectories or just store the averages.
 method: str, default=”adams”
Which ODE integrator methods are supported.
 map: str {“serial”, “parallel”, “loky”}
How to run the trajectories. “parallel” uses concurent module to run in parallel while “loky” use the module of the same name to do so.
 job_timeout: None, int
Maximum time to compute one trajectory.
 num_cpus: None, int
Number of cpus to use when running in parallel.
None
detect the number of available cpus. bitgenerator: {None, “MT19937”, “PCG64”, “PCG64DXSM”, …}
Which of numpy.random’s bitgenerator to use. With
None
, your numpy version’s default is used. mc_corr_eps: float
Small number used to detect nonphysical collapse caused by numerical imprecision.
 norm_t_tol: float
Tolerance in time used when finding the collapse.
 norm_tol: float
Tolerance in norm used when finding the collapse.
 norm_steps: int
Maximum number of tries to find the collapse.
 improved_sampling: Bool
Whether to use the improved sampling algorithm of Abdelhafez et al. PRA (2019)
 property resultclass
Base class for storing results for solver using multiple trajectories.
 Parameters:
 e_ops
Qobj
,QobjEvo
, function or list or dict of these The
e_ops
parameter defines the set of values to record at each time stept
. If an element is aQobj
orQobjEvo
the value recorded is the expectation value of that operator given the state att
. If the element is a function,f
, the value recorded isf(t, state)
.The values are recorded in the
.expect
attribute of this result object..expect
is a list, where each item contains the values of the correspondinge_op
.Function
e_ops
must return a number so the average can be computed. optionsdict
The options for this result class.
 solverstr or None
The name of the solver generating these results.
 statsdict or None
The stats generated by the solver while producing these results. Note that the solver may update the stats directly while producing results.
 kwdict
Additional parameters specific to a result subclass.
 e_ops
 run(state, tlist, ntraj=1, *, args=None, e_ops=(), timeout=None, target_tol=None, seed=None)[source]
Do the evolution of the Quantum system. See the overridden method for further details. The modification here is to sample the nojump trajectory first. Then, the nojump probability is used as a lowerbound for random numbers in future monte carlo runs
 start(state, t0, seed=None)
Set the initial state and time for a step evolution.
 Parameters:
 state
Qobj
Initial state of the evolution.
 t0double
Initial time of the evolution.
 seedint, SeedSequence, list, {None}
Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seed, one for each trajectory.
 ..note ::
When using step evolution, only one trajectory can be computed at once.
 state
 step(t, *, args=None, copy=True)
Evolve the state to
t
and return the state as aQobj
. Parameters:
 tdouble
Time to evolve to, must be higher than the last call.
 argsdict, optional {None}
Update the
args
of the system. The change is effective from the beginning of the interval. Changingargs
can slow the evolution. copybool, optional {True}
Whether to return a copy of the data or the data in the ODE solver.
 property sys_dims
Dimensions of the space that the system use:
qutip.basis(sovler.dims)
will create a state with proper dimensions for this solver.
 trajectory_resultclass
alias of
McTrajectoryResult
 class NonMarkovianMCSolver(H, ops_and_rates, *_args, args=None, options=None, **kwargs)[source]
Bases:
MCSolver
Monte Carlo Solver for Lindblad equations with “rates” that may be negative. The
c_ops
parameter ofqutip.MCSolver
is replaced by anops_and_rates
parameter to allow for negative rates. Options for the underlying ODE solver are given by the Options class. Parameters:
 H
qutip.Qobj
,qutip.QobjEvo
,list
, callable. System Hamiltonian as a Qobj, QobjEvo. It can also be any input type that QobjEvo accepts (see
qutip.QobjEvo
documentation).H
can also be a superoperator (liouvillian) if some collapse operators are to be treated deterministically. ops_and_rateslist
A
list
of tuples(L, Gamma)
, where the Lindblad operatorL
is aqutip.Qobj
andGamma
represents the corresponding rate, which is allowed to be negative. The Lindblad operators must be operators even ifH
is a superoperator. Each rateGamma
may be just a number (in the case of a constant rate) or, otherwise, specified using any format accepted byqutip.coefficient
. argsNone / dict
Arguments for timedependent Hamiltonian and collapse operator terms.
 optionsSolverOptions, [optional]
Options for the evolution.
 seedint, SeedSequence, list, [optional]
Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seed, one for each trajectory. Seeds are saved in the result and can be reused with:
seeds=prev_result.seeds
 H
 classmethod add_integrator(integrator, key)
Register an integrator.
 Parameters:
 integratorIntegrator
The ODE solver to register.
 keyslist of str
Values of the method options that refer to this integrator.
 current_martingale()[source]
Returns the value of the influence martingale along the current trajectory. The value of the martingale is the product of the continuous and the discrete contribution. The current time and the collapses that have happened are read out from the internal integrator.
 mc_integrator_class
alias of
NmMCIntegrator
 property options
Options for monte carlo solver:
 store_final_state: bool, default=False
Whether or not to store the final state of the evolution in the result class.
 store_states: bool, default=None
Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 progress_bar: str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, default=”text”
How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs: dict, default={“chunk_size”:10}
Arguments to pass to the progress_bar. Qutip’s bars use
chunk_size
. keep_runs_results: bool
Whether to store results from all trajectories or just store the averages.
 method: str, default=”adams”
Which ODE integrator methods are supported.
 map: str {“serial”, “parallel”, “loky”}
How to run the trajectories. “parallel” uses concurent module to run in parallel while “loky” use the module of the same name to do so.
 job_timeout: None, int
Maximum time to compute one trajectory.
 num_cpus: None, int
Number of cpus to use when running in parallel.
None
detect the number of available cpus. bitgenerator: {None, “MT19937”, “PCG64”, “PCG64DXSM”, …}
Which of numpy.random’s bitgenerator to use. With
None
, your numpy version’s default is used. mc_corr_eps: float
Small number used to detect nonphysical collapse caused by numerical imprecision.
 norm_t_tol: float
Tolerance in time used when finding the collapse.
 norm_tol: float
Tolerance in norm used when finding the collapse.
 norm_steps: int
Maximum number of tries to find the collapse.
 improved_sampling: Bool
Whether to use the improved sampling algorithm of Abdelhafez et al. PRA (2019)
 rate(t, i)[source]
Return the i’th unshifted rate at time
t
. Parameters:
 tfloat
The time at which to calculate the rate.
 iint
Which rate to calculate.
 Returns:
 ratefloat
The value of rate
i
at timet
.
 rate_shift(t)[source]
Return the rate shift at time
t
.The rate shift is
2 * abs(min([0, rate_1(t), rate_2(t), ...]))
. Parameters:
 tfloat
The time at which to calculate the rate shift.
 Returns:
 rate_shiftfloat
The rate shift amount.
 resultclass
alias of
NmmcResult
 run(state, tlist, *args, **kwargs)[source]
Do the evolution of the Quantum system.
For a
state
at timetlist[0]
do the evolution as directed byrhs
and for each time intlist
store the state and/or expectation values in aResult
. The evolution method and stored results are determined byoptions
. Parameters:
 state
Qobj
Initial state of the evolution.
 tlistlist of double
Time for which to save the results (state and/or expect) of the evolution. The first element of the list is the initial time of the evolution. Time in the list must be in increasing order, but does not need to be uniformly distributed.
 ntrajint
Number of trajectories to add.
 argsdict, optional {None}
Change the
args
of the rhs for the evolution. e_opslist
list of Qobj or QobjEvo to compute the expectation values. Alternatively, function[s] with the signature f(t, state) > expect can be used.
 timeoutfloat, optional [1e8]
Maximum time in seconds for the trajectories to run. Once this time is reached, the simulation will end even if the number of trajectories is less than
ntraj
. The map function, set in options, can interupt the running trajectory or wait for it to finish. Set to an arbitrary high number to disable. target_tol{float, tuple, list}, optional [None]
Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by
ntraj
. The error is computed using jackknife resampling.target_tol
can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of (atol, rtol) for each e_ops. seed{int, SeedSequence, list} optional
Seed or list of seeds for each trajectories.
 state
 Returns:
 results
qutip.solver.MultiTrajResult
Results of the evolution. States and/or expect will be saved. You can control the saved data in the options.
 results
 sqrt_shifted_rate(t, i)[source]
Return the square root of the i’th shifted rate at time
t
. Parameters:
 tfloat
The time at wich to calculate the shifted rate.
 iint
Which shifted rate to calculate.
 Returns:
 ratefloat
The square root of the shifted value of rate
i
at timet
.
 start(state, t0, seed=None)[source]
Set the initial state and time for a step evolution.
 Parameters:
 state
Qobj
Initial state of the evolution.
 t0double
Initial time of the evolution.
 seedint, SeedSequence, list, {None}
Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seed, one for each trajectory.
 ..note ::
When using step evolution, only one trajectory can be computed at once.
 state
 step(t, *, args=None, copy=True)[source]
Evolve the state to
t
and return the state as aQobj
. Parameters:
 tdouble
Time to evolve to, must be higher than the last call.
 argsdict, optional {None}
Update the
args
of the system. The change is effective from the beginning of the interval. Changingargs
can slow the evolution. copybool, optional {True}
Whether to return a copy of the data or the data in the ODE solver.
 property sys_dims
Dimensions of the space that the system use:
qutip.basis(sovler.dims)
will create a state with proper dimensions for this solver.
 trajectory_resultclass
alias of
NmmcTrajectoryResult
NonMarkovian HEOM Solver
 class HEOMSolver(H, bath, max_depth, *, options=None)[source]
HEOM solver that supports multiple baths.
The baths must be all either bosonic or fermionic baths.
 Parameters:
 H
Qobj
,QobjEvo
Possibly timedependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo. list of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. bathBath or list of Bath
A
Bath
containing the exponents of the expansion of the bath correlation funcion and their associated coefficients and coupling operators, or a list of baths.If multiple baths are given, they must all be either fermionic or bosonic baths.
 max_depthint
The maximum depth of the heirarchy (i.e. the maximum number of bath exponent “excitations” to retain).
 optionsdict, optional
Generic solver options. If set to None the default options will be used. Keyword only. Default: None.
 H
 Attributes:
 ados
HierarchyADOs
The description of the hierarchy constructed from the given bath and maximum depth.
 rhs
QobjEvo
The righthand side (RHS) of the hierarchy evolution ODE. Internally the system and bath coupling operators are converted to
qutip.data.CSR
instances during construction of the RHS, so the operators in therhs
will all be sparse.
 ados
 property options
Options for HEOMSolver:
 store_final_state: bool, default=False
Whether or not to store the final state of the evolution in the result class.
 store_states: bool, default=None
Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 normalize_output: bool, default=False
Normalize output state to hide ODE numerical errors.
 progress_bar: str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}, default=”text”
How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs: dict, default={“chunk_size”: 10}
Arguments to pass to the progress_bar. Qutip’s bars use
chunk_size
. method: str, default=”adams”
Which ordinary differential equation integration method to use.
 state_data_type: str, default=”dense”
Name of the data type of the state used during the ODE evolution. Use an empty string to keep the input state type. Many integrators support only work with Dense.
 store_adosbool, default=False
Whether or not to store the HEOM ADOs. Only relevant when using the HEOM solver.
 resultclass
alias of
HEOMResult
 run(state0, tlist, *, args=None, e_ops=None)[source]
Solve for the time evolution of the system.
 Parameters:
 state0
Qobj
orHierarchyADOsState
or arraylike If
rho0
is aQobj
the it is the initial state of the system (i.e. aQobj
density matrix).If it is a
HierarchyADOsState
or arraylike, thenrho0
gives the initial state of all ADOs.Usually the state of the ADOs would be determine from a previous call to
.run(...)
with the solver results optionstore_ados
set to True. For example,result = solver.run(...)
could be followed bysolver.run(result.ado_states[1], tlist)
.If a numpy arraylike is passed its shape must be
(number_of_ados, n, n)
where(n, n)
is the system shape (i.e. shape of the system density matrix) and the ADOs must be in the same order as in.ados.labels
. tlistlist
An ordered list of times at which to return the value of the state.
 argsdict, optional {None}
Change the
args
of the RHS for the evolution. e_opsQobj / QobjEvo / callable / list / dict / None, optional
A list or dictionary of operators as
Qobj
,QobjEvo
and/or callable functions (they can be mixed) or a single operator or callable function. For an operatorop
, the result will be computed using(state * op).tr()
and the state at each timet
. For callable functions,f
, the result is computed usingf(t, ado_state)
. The values are stored in theexpect
ande_data
attributes of the result (see the return section below).
 state0
 Returns:
HEOMResult
The results of the simulation run, with the following important attributes:
times
: the timest
(i.e. thetlist
).states
: the system state at each timet
(only available ife_ops
wasNone
or if the solver optionstore_states
was set toTrue
).ado_states
: the full ADO state at each time (only available if the results optionado_return
was set toTrue
). Each element is an instance ofHierarchyADOsState
. The state of a particular ADO may be extracted fromresult.ado_states[i]
by callingextract
.expect
: a list containing the values of eache_ops
at timet
.e_data
: a dictionary containing the values of eache_ops
at tmet
. The keys are those given bye_ops
if it was a dict, otherwise they are the indexes of the suppliede_ops
.
See
HEOMResult
andResult
for the complete list of attributes.
 start(state0, t0)[source]
Set the initial state and time for a step evolution.
 Parameters:
 state0
Qobj
Initial state of the evolution. This may provide either just the initial density matrix of the system, or the full set of ADOs for the hierarchy. See the documentation for
rho0
in the.run(...)
method for details. t0double
Initial time of the evolution.
 state0
 steady_state(use_mkl=True, mkl_max_iter_refine=100, mkl_weighted_matching=False)[source]
Compute the steady state of the system.
 Parameters:
 use_mklbool, default=False
Whether to use mkl or not. If mkl is not installed or if this is false, use the scipy splu solver instead.
 mkl_max_iter_refineint
Specifies the the maximum number of iterative refinement steps that the MKL PARDISO solver performs.
For a complete description, see iparm(8) in http://cali2.unilim.fr/intelxe/mkl/mklman/GUID264E311EACED4D56AC31E9D3B11D1CBF.htm.
 mkl_weighted_matchingbool
MKL PARDISO can use a maximum weighted matching algorithm to permute large elements close the diagonal. This strategy adds an additional level of reliability to the factorization methods.
For a complete description, see iparm(13) in http://cali2.unilim.fr/intelxe/mkl/mklman/GUID264E311EACED4D56AC31E9D3B11D1CBF.htm.
 Returns:
 steady_stateQobj
The steady state density matrix of the system.
 steady_ados
HierarchyADOsState
The steady state of the full ADO hierarchy. A particular ADO may be extracted from the full state by calling
HEOMSolver.extract
.
 property sys_dims
Dimensions of the space that the system use, excluding any environment:
qutip.basis(sovler.dims)
will create a state with proper dimensions for this solver.
 class HSolverDL(H_sys, coup_op, coup_strength, temperature, N_cut, N_exp, cut_freq, *, bnd_cut_approx=False, options=None, combine=True)[source]
A helper class for creating an
HEOMSolver
that is backwards compatible with theHSolverDL
provided inqutip.nonmarkov.heom
in QuTiP 4.6 and below.See
HEOMSolver
andDrudeLorentzBath
for more descriptions of the underlying solver and bath construction.An exact copy of the QuTiP 4.6 HSolverDL is provided in
qutip.nonmarkov.dlheom_solver
for cases where the functionality of the older solver is required. The older solver will be completely removed in QuTiP 5.Note
Unlike the version of
HSolverDL
in QuTiP 4.6, this solver supports supplying a timedependent or LiouvillianH_sys
.Note
For compatibility with
HSolverDL
in QuTiP 4.6 and below, the parameterN_exp
specifying the number of exponents to keep in the expansion of the bath correlation function is one more than the equivalentNk
used in theDrudeLorentzBath
. I.e.,Nk = N_exp  1
. TheNk
parameter in theDrudeLorentzBath
does not count the zeroeth exponent in order to better match common usage in the literature.Note
The
stats
andrenorm
arguments accepted in QuTiP 4.6 and below are no longer supported. Parameters:
 H_sysQobj or QobjEvo or list
The system Hamiltonian or Liouvillian. See
HEOMSolver
for a complete description. coup_opQobj
Operator describing the coupling between system and bath. See parameter
Q
inBosonicBath
for a complete description. coup_strengthfloat
Coupling strength. Referred to as
lam
inDrudeLorentzBath
. temperaturefloat
Bath temperature. Referred to as
T
inDrudeLorentzBath
. N_cutint
The maximum depth of the hierarchy. See
max_depth
inHEOMSolver
for a full description. N_expint
Number of exponential terms used to approximate the bath correlation functions. The equivalent
Nk
inDrudeLorentzBath
is one less thanN_exp
(see note above). cut_freqfloat
Bath spectral density cutoff frequency. Referred to as
gamma
inDrudeLorentzBath
. bnd_cut_approxbool
Use boundary cut off approximation. If true, the Matsubara terminator is added to the system Liouvillian (and H_sys is promoted to a Liouvillian if it was a Hamiltonian). Keyword only. Default: False.
 optionsdict, optional
Generic solver options. If set to None the default options will be used. Keyword only. Default: None.
 combinebool, default True
Whether to combine exponents with the same frequency (and coupling operator). See
BosonicBath.combine
for details. Keyword only. Default: True.
 class BathExponent(type, dim, Q, ck, vk, ck2=None, sigma_bar_k_offset=None, tag=None)[source]
Represents a single exponent (naively, an excitation mode) within the decomposition of the correlation functions of a bath.
 Parameters:
 type{“R”, “I”, “RI”, “+”, “”} or BathExponent.ExponentType
The type of bath exponent.
“R” and “I” are bosonic bath exponents that appear in the real and imaginary parts of the correlation expansion.
“RI” is combined bosonic bath exponent that appears in both the real and imaginary parts of the correlation expansion. The combined exponent has a single
vk
. Theck
is the coefficient in the real expansion andck2
is the coefficient in the imaginary expansion.“+” and “” are fermionic bath exponents. These fermionic bath exponents must specify
sigma_bar_k_offset
which specifies the amount to add tok
(the exponent index within the bath of this exponent) to determine thek
of the corresponding exponent with the opposite sign (i.e. “” or “+”). dimint or None
The dimension (i.e. maximum number of excitations for this exponent). Usually
2
for fermionic exponents orNone
(i.e. unlimited) for bosonic exponents. QQobj
The coupling operator for this excitation mode.
 vkcomplex
The frequency of the exponent of the excitation term.
 ckcomplex
The coefficient of the excitation term.
 ck2optional, complex
For exponents of type “RI” this is the coefficient of the term in the imaginary expansion (and
ck
is the coefficient in the real expansion). sigma_bar_k_offsetoptional, int
For exponents of type “+” this gives the offset (within the list of exponents within the bath) of the corresponding “” bath exponent. For exponents of type “” it gives the offset of the corresponding “+” exponent.
 tagoptional, str, tuple or any other object
A label for the exponent (often the name of the bath). It defaults to None.
 Attributes:
 fermionicbool
True if the type of the exponent is a Fermionic type (i.e. either “+” or “”) and False otherwise.
 All of the parameters are also available as attributes.
 types
alias of
ExponentType
 class Bath(exponents)[source]
Represents a list of bath expansion exponents.
 Parameters:
 exponentslist of BathExponent
The exponents of the correlation function describing the bath.
 Attributes:
 All of the parameters are available as attributes.
 class BosonicBath(Q, ck_real, vk_real, ck_imag, vk_imag, combine=True, tag=None)[source]
A helper class for constructing a bosonic bath from the expansion coefficients and frequencies for the real and imaginary parts of the bath correlation function.
If the correlation functions
C(t)
is split into real and imaginary parts:C(t) = C_real(t) + i * C_imag(t)
then:
C_real(t) = sum(ck_real * exp( vk_real * t)) C_imag(t) = sum(ck_imag * exp( vk_imag * t))
Defines the coefficients
ck
and the frequenciesvk
.Note that the
ck
andvk
may be complex, even throughC_real(t)
andC_imag(t)
(i.e. the sum) is real. Parameters:
 QQobj
The coupling operator for the bath.
 ck_reallist of complex
The coefficients of the expansion terms for the real part of the correlation function. The corresponding frequencies are passed as vk_real.
 vk_reallist of complex
The frequencies (exponents) of the expansion terms for the real part of the correlation function. The corresponding ceofficients are passed as ck_real.
 ck_imaglist of complex
The coefficients of the expansion terms in the imaginary part of the correlation function. The corresponding frequencies are passed as vk_imag.
 vk_imaglist of complex
The frequencies (exponents) of the expansion terms for the imaginary part of the correlation function. The corresponding ceofficients are passed as ck_imag.
 combinebool, default True
Whether to combine exponents with the same frequency (and coupling operator). See
combine
for details. tagoptional, str, tuple or any other object
A label for the bath exponents (for example, the name of the bath). It defaults to None but can be set to help identify which bath an exponent is from.
 classmethod combine(exponents, rtol=1e05, atol=1e07)[source]
Group bosonic exponents with the same frequency and return a single exponent for each frequency present.
Exponents with the same frequency are only combined if they share the same coupling operator
.Q
.Note that combined exponents take their tag from the first exponent in the group being combined (i.e. the one that occurs first in the given exponents list).
 Parameters:
 exponentslist of BathExponent
The list of exponents to combine.
 rtolfloat, default 1e5
The relative tolerance to use to when comparing frequencies and coupling operators.
 atolfloat, default 1e7
The absolute tolerance to use to when comparing frequencies and coupling operators.
 Returns:
 list of BathExponent
The new reduced list of exponents.
 class DrudeLorentzBath(Q, lam, gamma, T, Nk, combine=True, tag=None)[source]
A helper class for constructing a DrudeLorentz bosonic bath from the bath parameters (see parameters below).
 Parameters:
 QQobj
Operator describing the coupling between system and bath.
 lamfloat
Coupling strength.
 gammafloat
Bath spectral density cutoff frequency.
 Tfloat
Bath temperature.
 Nkint
Number of exponential terms used to approximate the bath correlation functions.
 combinebool, default True
Whether to combine exponents with the same frequency (and coupling operator). See
BosonicBath.combine
for details. tagoptional, str, tuple or any other object
A label for the bath exponents (for example, the name of the bath). It defaults to None but can be set to help identify which bath an exponent is from.
 terminator()[source]
Return the Matsubara terminator for the bath and the calculated approximation discrepancy.
 Returns:
 delta: float
The approximation discrepancy. That is, the difference between the true correlation function of the DrudeLorentz bath and the sum of the
Nk
exponential terms is approximately2 * delta * dirac(t)
, wheredirac(t)
denotes the Dirac delta function. terminatorQobj
The Matsubara terminator – i.e. a liouvillian term representing the contribution to the systembath dynamics of all exponential expansion terms beyond
Nk
. It should be used by adding it to the system liouvillian (i.e.liouvillian(H_sys)
).
 class DrudeLorentzPadeBath(Q, lam, gamma, T, Nk, combine=True, tag=None)[source]
A helper class for constructing a Padé expansion for a DrudeLorentz bosonic bath from the bath parameters (see parameters below).
A Padé approximant is a sumoverpoles expansion ( see https://en.wikipedia.org/wiki/Pad%C3%A9_approximant).
The application of the Padé method to spectrum decompoisitions is described in “Padé spectrum decompositions of quantum distribution functions and optimal hierarchical equations of motion construction for quantum open systems” [1].
The implementation here follows the approach in the paper.
[1] J. Chem. Phys. 134, 244106 (2011); https://doi.org/10.1063/1.3602466
This is an alternative to the
DrudeLorentzBath
which constructs a simpler exponential expansion. Parameters:
 QQobj
Operator describing the coupling between system and bath.
 lamfloat
Coupling strength.
 gammafloat
Bath spectral density cutoff frequency.
 Tfloat
Bath temperature.
 Nkint
Number of Padé exponentials terms used to approximate the bath correlation functions.
 combinebool, default True
Whether to combine exponents with the same frequency (and coupling operator). See
BosonicBath.combine
for details. tagoptional, str, tuple or any other object
A label for the bath exponents (for example, the name of the bath). It defaults to None but can be set to help identify which bath an exponent is from.
 terminator()[source]
Return the Padé terminator for the bath and the calculated approximation discrepancy.
 Returns:
 delta: float
The approximation discrepancy. That is, the difference between the true correlation function of the DrudeLorentz bath and the sum of the
Nk
exponential terms is approximately2 * delta * dirac(t)
, wheredirac(t)
denotes the Dirac delta function. terminatorQobj
The Padé terminator – i.e. a liouvillian term representing the contribution to the systembath dynamics of all exponential expansion terms beyond
Nk
. It should be used by adding it to the system liouvillian (i.e.liouvillian(H_sys)
).
 class UnderDampedBath(Q, lam, gamma, w0, T, Nk, combine=True, tag=None)[source]
A helper class for constructing an underdamped bosonic bath from the bath parameters (see parameters below).
 Parameters:
 QQobj
Operator describing the coupling between system and bath.
 lamfloat
Coupling strength.
 gammafloat
Bath spectral density cutoff frequency.
 w0float
Bath spectral density resonance frequency.
 Tfloat
Bath temperature.
 Nkint
Number of exponential terms used to approximate the bath correlation functions.
 combinebool, default True
Whether to combine exponents with the same frequency (and coupling operator). See
BosonicBath.combine
for details. tagoptional, str, tuple or any other object
A label for the bath exponents (for example, the name of the bath). It defaults to None but can be set to help identify which bath an exponent is from.
 class FermionicBath(Q, ck_plus, vk_plus, ck_minus, vk_minus, tag=None)[source]
A helper class for constructing a fermionic bath from the expansion coefficients and frequencies for the
+
and
modes of the bath correlation function.There must be the same number of
+
and
modes and their coefficients must be specified in the same order so thatck_plus[i], vk_plus[i]
are the plus coefficient and frequency corresponding to the minus modeck_minus[i], vk_minus[i]
.In the fermionic case the order in which excitations are created or destroyed is important, resulting in two different correlation functions labelled
C_plus(t)
andC_plus(t)
:C_plus(t) = sum(ck_plus * exp( vk_plus * t)) C_minus(t) = sum(ck_minus * exp( vk_minus * t))
where the expansions above define the coeffiients
ck
and the frequenciesvk
. Parameters:
 QQobj
The coupling operator for the bath.
 ck_pluslist of complex
The coefficients of the expansion terms for the
+
part of the correlation function. The corresponding frequencies are passed as vk_plus. vk_pluslist of complex
The frequencies (exponents) of the expansion terms for the
+
part of the correlation function. The corresponding ceofficients are passed as ck_plus. ck_minuslist of complex
The coefficients of the expansion terms for the

part of the correlation function. The corresponding frequencies are passed as vk_minus. vk_minuslist of complex
The frequencies (exponents) of the expansion terms for the

part of the correlation function. The corresponding ceofficients are passed as ck_minus. tagoptional, str, tuple or any other object
A label for the bath exponents (for example, the name of the bath). It defaults to None but can be set to help identify which bath an exponent is from.
 class LorentzianBath(Q, gamma, w, mu, T, Nk, tag=None)[source]
A helper class for constructing a Lorentzian fermionic bath from the bath parameters (see parameters below).
Note
This Matsubara expansion used in this bath converges very slowly and
Nk > 20
may be required to get good convergence. The Padé expansion used byLorentzianPadeBath
converges much more quickly. Parameters:
 QQobj
Operator describing the coupling between system and bath.
 gammafloat
The coupling strength between the system and the bath.
 wfloat
The width of the environment.
 mufloat
The chemical potential of the bath.
 Tfloat
Bath temperature.
 Nkint
Number of exponential terms used to approximate the bath correlation functions.
 tagoptional, str, tuple or any other object
A label for the bath exponents (for example, the name of the bath). It defaults to None but can be set to help identify which bath an exponent is from.
 class LorentzianPadeBath(Q, gamma, w, mu, T, Nk, tag=None)[source]
A helper class for constructing a Padé expansion for Lorentzian fermionic bath from the bath parameters (see parameters below).
A Padé approximant is a sumoverpoles expansion ( see https://en.wikipedia.org/wiki/Pad%C3%A9_approximant).
The application of the Padé method to spectrum decompoisitions is described in “Padé spectrum decompositions of quantum distribution functions and optimal hierarchical equations of motion construction for quantum open systems” [1].
The implementation here follows the approach in the paper.
[1] J. Chem. Phys. 134, 244106 (2011); https://doi.org/10.1063/1.3602466
This is an alternative to the
LorentzianBath
which constructs a simpler exponential expansion that converges much more slowly in this particular case. Parameters:
 QQobj
Operator describing the coupling between system and bath.
 gammafloat
The coupling strength between the system and the bath.
 wfloat
The width of the environment.
 mufloat
The chemical potential of the bath.
 Tfloat
Bath temperature.
 Nkint
Number of exponential terms used to approximate the bath correlation functions.
 tagoptional, str, tuple or any other object
A label for the bath exponents (for example, the name of the bath). It defaults to None but can be set to help identify which bath an exponent is from.
 class HierarchyADOs(exponents, max_depth)[source]
A description of ADOs (auxilliary density operators) with the hierarchical equations of motion.
The list of ADOs is constructed from a list of bath exponents (corresponding to one or more baths). Each ADO is referred to by a label that lists the number of “excitations” of each bath exponent. The level of a label within the hierarchy is the sum of the “excitations” within the label.
For example the label
(0, 0, ..., 0)
represents the density matrix of the system being solved and is the only 0th level label.The labels with a single 1, i.e.
(1, 0, ..., 0)
,(0, 1, 0, ... 0)
, etc. are the 1st level labels.The second level labels all have either two 1s or a single 2, and so on for the third and higher levels of the hierarchy.
 Parameters:
 exponentslist of BathExponent
The exponents of the correlation function describing the bath or baths.
 max_depthint
The maximum depth of the hierarchy (i.e. the maximum sum of “excitations” in the hierarchy ADO labels or maximum ADO level).
 Attributes:
 exponentslist of BathExponent
The exponents of the correlation function describing the bath or baths.
 max_depthint
The maximum depth of the hierarchy (i.e. the maximum sum of “excitations” in the hierarchy ADO labels).
 dimslist of int
The dimensions of each exponent within the bath(s).
 vklist of complex
The frequency of each exponent within the bath(s).
 cklist of complex
The coefficient of each exponent within the bath(s).
 ck2: list of complex
For exponents of type “RI”, the coefficient of the exponent within the imaginary expansion. For other exponent types, the entry is None.
 sigma_bar_k_offset: list of int
For exponents of type “+” or “” the offset within the list of modes of the corresponding “” or “+” exponent. For other exponent types, the entry is None.
 labels: list of tuples
A list of the ADO labels within the hierarchy.
 exps(label)[source]
Converts an ADO label into a tuple of exponents, with one exponent for each “excitation” within the label.
The number of exponents returned is always equal to the level of the label within the hierarchy (i.e. the sum of the indices within the label).
 Parameters:
 labeltuple
The ADO label to convert to a list of exponents.
 Returns:
 tuple of BathExponent
A tuple of BathExponents.
Examples
ados.exps((1, 0, 0))
would return[ados.exponents[0]]
ados.exps((2, 0, 0))
would return[ados.exponents[0], ados.exponents[0]]
.ados.exps((1, 2, 1))
would return[ados.exponents[0], ados.exponents[1], ados.exponents[1], ados.exponents[2]]
.
 filter(level=None, tags=None, dims=None, types=None)[source]
Return a list of ADO labels for ADOs whose “excitations” match the given patterns.
Each of the filter parameters (tags, dims, types) may be either unspecified (None) or a list. Unspecified parameters are excluded from the filtering.
All specified filter parameters must be lists of the same length. Each position in the lists describes a particular excitation and any exponent that matches the filters may supply that excitation. The level of all labels returned is thus equal to the length of the filter parameter lists.
Within a filter parameter list, items that are None represent wildcards and match any value of that exponent attribute
 Parameters:
 levelint
The hierarchy depth to return ADOs from.
 tagslist of object or None
Filter parameter that matches the
.tag
attribute of exponents. dimslist of int
Filter parameter that matches the
.dim
attribute of exponents. typeslist of BathExponent types or list of str
Filter parameter that matches the
.type
attribute of exponents. Types may be supplied by name (e.g. “R”, “I”, “+”) instead of by the actual type (e.g.BathExponent.types.R
).
 Returns:
 list of tuple
The ADO label for each ADO whose exponent excitations (i.e. label) match the given filters or level.
 idx(label)[source]
Return the index of the ADO label within the list of labels, i.e. within
self.labels
. Parameters:
 labeltuple
The label to look up.
 Returns:
 int
The index of the label within the list of ADO labels.
 next(label, k)[source]
Return the ADO label with one more excitation in the k’th exponent dimension or
None
if adding the excitation would exceed the dimension or maximum depth of the hierarchy. Parameters:
 labeltuple
The ADO label to add an excitation to.
 kint
The exponent to add the excitation to.
 Returns:
 tuple or None
The next label.
 prev(label, k)[source]
Return the ADO label with one fewer excitation in the k’th exponent dimension or
None
if the label has no exciations in the k’th exponent. Parameters:
 labeltuple
The ADO label to remove the excitation from.
 kint
The exponent to remove the excitation from.
 Returns:
 tuple or None
The previous label.
 class HierarchyADOsState(rho, ados, ado_state)[source]
Provides convenient access to the full hierarchy ADO state at a particular point in time,
t
. Parameters:
 rho
Qobj
The current state of the system (i.e. the 0th component of the hierarchy).
 ados
HierarchyADOs
The description of the hierarchy.
 ado_statenumpy.array
The full state of the hierarchy.
 rho
 Attributes:
 rhoQobj
The system state.
 In addition, all of the attributes of the hierarchy description,
 i.e. ``HierarchyADOs``, are provided directly on this class for
 convenience. E.g. one can access ``.labels``, or ``.exponents`` or
 call ``.idx(label)`` directly.
 See :class:`HierarchyADOs` for a full list of the available attributes
 and methods.
 extract(idx_or_label)[source]
Extract a Qobj representing the specified ADO from a full representation of the ADO states.
 Parameters:
 idxint or label
The index of the ADO to extract. If an ADO label, e.g.
(0, 1, 0, ...)
is supplied instead, then the ADO is extracted by label instead.
 Returns:
 Qobj
A
Qobj
representing the state of the specified ADO.
Integrator
 class IntegratorScipyAdams(system, options)[source]
Integrator using Scipy ode with zvode integrator using adams method. Ordinary Differential Equation solver by netlib (http://www.netlib.org/odepack).
Usable with
method="adams"
 property options
Supported options by zvode integrator:
 atolfloat, default=1e8
Absolute tolerance.
 rtolfloat, default=1e6
Relative tolerance.
 orderint, default=12, ‘adams’ or 5, ‘bdf’
Order of integrator <=12 ‘adams’, <=5 ‘bdf’
 nstepsint, default=2500
Max. number of internal steps/call.
 first_stepfloat, default=0
Size of initial step (0 = automatic).
 min_stepfloat, default=0
Minimum step size (0 = automatic).
 max_stepfloat, default=0
Maximum step size (0 = automatic) When using pulses, change to half the thinest pulse otherwise it may be skipped.
 class IntegratorScipyBDF(system, options)[source]
Integrator using Scipy ode with zvode integrator using bdf method. Ordinary Differential Equation solver by netlib (http://www.netlib.org/odepack).
Usable with
method="bdf"
 property options
Supported options by zvode integrator:
 atolfloat, default=1e8
Absolute tolerance.
 rtolfloat, default=1e6
Relative tolerance.
 orderint, default=12, ‘adams’ or 5, ‘bdf’
Order of integrator <=12 ‘adams’, <=5 ‘bdf’
 nstepsint, default=2500
Max. number of internal steps/call.
 first_stepfloat, default=0
Size of initial step (0 = automatic).
 min_stepfloat, default=0
Minimum step size (0 = automatic).
 max_stepfloat, default=0
Maximum step size (0 = automatic) When using pulses, change to half the thinest pulse otherwise it may be skipped.
 class IntegratorScipylsoda(system, options)[source]
Integrator using Scipy ode with lsoda integrator. ODE solver by netlib (http://www.netlib.org/odepack) Automatically choose between ‘Adams’ and ‘BDF’ methods to solve both stiff and nonstiff systems.
Usable with
method="lsoda"
 property options
Supported options by lsoda integrator:
 atolfloat, default=1e8
Absolute tolerance.
 rtolfloat, default=1e6
Relative tolerance.
 nstepsint, default=2500
Max. number of internal steps/call.
 max_order_nsint, default=12
Maximum order used in the nonstiff case (<= 12).
 max_order_sint, default=5
Maximum order used in the stiff case (<= 5).
 first_stepfloat, default=0
Size of initial step (0 = automatic).
 max_stepfloat, default=0
Maximum step size (0 = automatic) When using pulses, change to half the thinest pulse otherwise it may be skipped.
 min_stepfloat, default=0
Minimum step size (0 = automatic)
 class IntegratorScipyDop853(system, options)[source]
Integrator using Scipy ode with dop853 integrator. Eight order rungekutta method by Dormand & Prince. Use fortran implementation from [E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary Differential Equations i. Nonstiff Problems. 2nd edition. Springer Series in Computational Mathematics, SpringerVerlag (1993)].
Usable with
method="dop853"
 property options
Supported options by dop853 integrator:
 atolfloat, default=1e8
Absolute tolerance.
 rtolfloat, default=1e6
Relative tolerance.
 nstepsint, default=2500
Max. number of internal steps/call.
 first_stepfloat, default=0
Size of initial step (0 = automatic).
 max_stepfloat, default=0
Maximum step size (0 = automatic)
 ifactor, dfactorfloat, default=6., 0.3
Maximum factor to increase/decrease step size by in one step
 betafloat, default=0
Beta parameter for stabilised step size control.
See scipy.integrate.ode ode for more detail
 class IntegratorVern7(system, options)[source]
QuTiP’s implementation of Verner’s “most efficient” RungeKutta method of order 7. These are RungeKutta methods with variable steps and dense output.
The implementation uses QuTiP’s Data objects for the state, allowing sparse, GPU or other data layer objects to be used efficiently by the solver in their native formats.
See http://people.math.sfu.ca/~jverner/ for a detailed description of the methods.
Usable with
method="vern7"
 property options
Supported options by verner method:
 atolfloat, default=1e8
Absolute tolerance.
 rtolfloat, default=1e6
Relative tolerance.
 nstepsint, default=1000
Max. number of internal steps/call.
 first_stepfloat, default=0
Size of initial step (0 = automatic).
 min_stepfloat, default=0
Minimum step size (0 = automatic).
 max_stepfloat, default=0
Maximum step size (0 = automatic) When using pulses, change to half the thinest pulse otherwise it may be skipped.
 interpolatebool, default=True
Whether to use interpolation step, faster most of the time.
 class IntegratorVern9(system, options)[source]
QuTiP’s implementation of Verner’s “most efficient” RungeKutta method of order 9. These are RungeKutta methods with variable steps and dense output.
The implementation uses QuTiP’s Data objects for the state, allowing sparse, GPU or other data layer objects to be used efficiently by the solver in their native formats.
See http://people.math.sfu.ca/~jverner/ for a detailed description of the methods.
Usable with
method="vern9"
 property options
Supported options by verner method:
 atolfloat, default=1e8
Absolute tolerance.
 rtolfloat, default=1e6
Relative tolerance.
 nstepsint, default=1000
Max. number of internal steps/call.
 first_stepfloat, default=0
Size of initial step (0 = automatic).
 min_stepfloat, default=0
Minimum step size (0 = automatic).
 max_stepfloat, default=0
Maximum step size (0 = automatic) When using pulses, change to half the thinest pulse otherwise it may be skipped.
 interpolatebool, default=True
Whether to use interpolation step, faster most of the time.
 class IntegratorDiag(system, options)[source]
Integrator solving the ODE by diagonalizing the system and solving analytically. It can only solve constant system and has a long preparation time, but the integration is fast.
Usable with
method="diag"
 property options
Supported options by “diag” method:
 eigensolver_dtypestr, default=”dense”
Qutip data type {“dense”, “csr”, etc.} to use when computing the eigenstates. The dense eigen solver is usually faster and more stable.
 class IntegratorKrylov(system, options)[source]
Evolve the state vector (“psi0”) finding an approximation for the time evolution operator of Hamiltonian (“H”) by obtaining the projection of the time evolution operator on a set of small dimensional Krylov subspaces (m << dim(H)).
 property options
Supported options by krylov method:
 atolfloat, default=1e7
Absolute tolerance.
 nstepsint, default=100
Max. number of internal steps/call.
 min_step, max_stepfloat, default=(1e5, 1e5)
Minimum and maximum step size.
 krylov_dim: int, default=0
Dimension of Krylov approximation subspaces used for the time evolution approximation. If the defaut 0 is given, the dimension is calculated from the system size N, using min(int((N + 100)**0.5), N1).
 sub_system_tol: float, default=1e7
Tolerance to detect a happy breakdown. A happy breakdown occurs when the initial ket is in a subspace of the Hamiltonian smaller than
krylov_dim
. always_compute_step: bool, default=False
If True, the step length is computed each time a new Krylov subspace is computed. Otherwise it is computed only once when creating the integrator.
Stochastic Integrator
 class RouchonSODE(rhs, options)[source]
Stochastic integration method keeping the positivity of the density matrix. See eq. (4) Pierre Rouchon and Jason F. Ralpha, Efficient Quantum Filtering for Quantum Feedback Control, arXiv:1410.5345 [quantph], Phys. Rev. A 91, 012118, (2015).
Order: strong 1
Note
This method should be used with very small
dt
. Unlike other methods that will return unphysical state (negative eigenvalues, Nans) when the time step is too large, this method will return state that seems normal. property options
Supported options by Rouchon Stochastic Integrators:
 dtfloat, default=0.001
Internal time step.
 tolfloat, default=1e7
Relative tolerance.
 class EulerSODE(rhs, options)[source]
A simple generalization of the Euler method for ordinary differential equations to stochastic differential equations. Only solver which could take noncommuting
sc_ops
.Order: 0.5
 property options
Supported options by Explicit Stochastic Integrators:
 dtfloat, default=0.001
Internal time step.
 tolfloat, default=1e10
Tolerance for the time steps.
 class Milstein_SODE(rhs, options)[source]
An order 1.0 strong Taylor scheme. Better approximate numerical solution to stochastic differential equations. See eq. (3.12) of chapter 10.3 of Peter E. Kloeden and Exkhard Platen, Numerical Solution of Stochastic Differential Equations..
Order strong 1.0
 property options
Supported options by Explicit Stochastic Integrators:
 dtfloat, default=0.001
Internal time step.
 tolfloat, default=1e10
Tolerance for the time steps.
 class Taylor1_5_SODE(rhs, options)[source]
Order 1.5 strong Taylor scheme. Solver with more terms of the ItoTaylor expansion. See eq. (4.6) of chapter 10.4 of Peter E. Kloeden and Exkhard Platen, Numerical Solution of Stochastic Differential Equations.
Order strong 1.5
 property options
Supported options by Order 1.5 strong Taylor Stochastic Integrators:
 dtfloat, default=0.001
Internal time step.
 tolfloat, default=1e10
Relative tolerance.
 derr_dtfloat, default=1e6
Finite time difference used to compute the derrivative of the hamiltonian and
sc_ops
.
 class Implicit_Milstein_SODE(rhs, options)[source]
An order 1.0 implicit strong Taylor scheme. Implicit Milstein scheme for the numerical simulation of stiff stochastic differential equations. Eq. (2.11) with alpha=0.5 of chapter 12.2 of Peter E. Kloeden and Exkhard Platen, Numerical Solution of Stochastic Differential Equations.
Order strong 1.0
 property options
Supported options by Implicit Stochastic Integrators:
 dtfloat, default=0.001
Internal time step.
 tolfloat, default=1e10
Tolerance for the time steps.
 solve_methodstr, default=None
Method used for solver the
Ax=b
of the implicit step. Accept methods supported byqutip.core.data.solve
. When the system is constant, the inverse of the matrixA
can be used by enteringinv
. solve_optionsdict, default={}
Options to pass to the call to
qutip.core.data.solve
.
 class Implicit_Taylor1_5_SODE(rhs, options)[source]
Order 1.5 implicit strong Taylor scheme. Solver with more terms of the ItoTaylor expansion. Eq. (2.18) with
alpha=0.5
of chapter 12.2 of Peter E. Kloeden and Exkhard Platen, Numerical Solution of Stochastic Differential Equations.Order strong 1.5
 property options
Supported options by Implicit Order 1.5 strong Taylor Stochastic Integrators:
 dtfloat, default=0.001
Internal time step.
 tolfloat, default=1e10
Tolerance for the time steps.
 solve_methodstr, default=None
Method used for solver the
Ax=b
of the implicit step. Accept methods supported byqutip.core.data.solve
. When the system is constant, the inverse of the matrixA
can be used by enteringinv
. solve_optionsdict, default={}
Options to pass to the call to
qutip.core.data.solve
. derr_dtfloat, default=1e6
Finite time difference used to compute the derrivative of the hamiltonian and
sc_ops
.
 class PlatenSODE(rhs, options)[source]
Explicit scheme, creates the Milstein using finite differences instead of analytic derivatives. Also contains some higher order terms, thus converges better than Milstein while staying strong order 1.0. Does not require derivatives. See eq. (7.47) of chapter 7 of H.P. Breuer and F. Petruccione, The Theory of Open Quantum Systems.
Order: strong 1, weak 2
 property options
Supported options by Explicit Stochastic Integrators:
 dtfloat, default=0.001
Internal time step.
 tolfloat, default=1e10
Tolerance for the time steps.
 class Explicit1_5_SODE(rhs, options)[source]
Explicit order 1.5 strong schemes. Reproduce the order 1.5 strong Taylor scheme using finite difference instead of derivatives. Slower than
taylor15
but usable when derrivatives cannot be analytically obtained. See eq. (2.13) of chapter 11.2 of Peter E. Kloeden and Exkhard Platen, Numerical Solution of Stochastic Differential Equations.Order: strong 1.5
 property options
Supported options by Explicit Stochastic Integrators:
 dtfloat, default=0.001
Internal time step.
 tolfloat, default=1e10
Tolerance for the time steps.
 class PredCorr_SODE(rhs, options)[source]
Generalization of the trapezoidal method to stochastic differential equations. More stable than explicit methods. See eq. (5.4) of chapter 15.5 of Peter E. Kloeden and Exkhard Platen, Numerical Solution of Stochastic Differential Equations.
Order strong 0.5, weak 1.0
Codes to only correct the stochastic part (\(\alpha=0\), \(\eta=1/2\)):
'predcorr'
,'predictorcorrector'
or'pceuler'
Codes to correct both the stochastic and deterministic parts (\(\alpha=1/2\), \(\eta=1/2\)):
'pceulerimp'
,'pceuler2'
or'predcorr2'
 property options
Supported options by Explicit Stochastic Integrators:
 dtfloat, default=0.001
Internal time step.
 tolfloat, default=1e10
Tolerance for the time steps.
 alphafloat, default=0.
Implicit factor to the drift. eff_drift ~= drift(t) * (1alpha) + drift(t+dt) * alpha
 etafloat, default=0.5
Implicit factor to the diffusion. eff_diffusion ~= diffusion(t) * (1eta) + diffusion(t+dt) * eta
Solver Options and Results
 class Result(e_ops, options, *, solver=None, stats=None, **kw)[source]
Base class for storing solver results.
 Parameters:
 e_ops
Qobj
,QobjEvo
, function or list or dict of these The
e_ops
parameter defines the set of values to record at each time stept
. If an element is aQobj
orQobjEvo
the value recorded is the expectation value of that operator given the state att
. If the element is a function,f
, the value recorded isf(t, state)
.The values are recorded in the
e_data
andexpect
attributes of this result object.e_data
is a dictionary andexpect
is a list, where each item contains the values of the correspondinge_op
. optionsdict
The options for this result class.
 solverstr or None
The name of the solver generating these results.
 statsdict or None
The stats generated by the solver while producing these results. Note that the solver may update the stats directly while producing results.
 kwdict
Additional parameters specific to a result subclass.
 e_ops
 Attributes:
 timeslist
A list of the times at which the expectation values and states were recorded.
 stateslist of
Qobj
The state at each time
t
(if the recording of the state was requested). final_state
Qobj
: The final state (if the recording of the final state was requested).
 expectlist of arrays of expectation values
A list containing the values of each
e_op
. The list is in the same order in which thee_ops
were supplied and empty if noe_ops
were given.Each element is itself a list and contains the values of the corresponding
e_op
, with one value for each time in.times
.The same lists of values may be accessed via the
.e_data
dictionary and the originale_ops
are available via the.e_ops
attribute. e_datadict
A dictionary containing the values of each
e_op
. If thee_ops
were supplied as a dictionary, the keys are the same as in that dictionary. Otherwise the keys are the index of thee_op
in the.expect
list.The lists of expectation values returned are the same lists as those returned by
.expect
. e_opsdict
A dictionary containing the supplied e_ops as
ExpectOp
instances. The keys of the dictionary are the same as for.e_data
. Each value is object where.e_ops[k](t, state)
calculates the value ofe_op
k
at timet
and the givenstate
, and.e_ops[k].op
is the original object supplied to create thee_op
. solverstr or None
The name of the solver generating these results.
 statsdict or None
The stats generated by the solver while producing these results.
 optionsdict
The options for this result class.
 add(t, state)[source]
Add a state to the results for the time
t
of the evolution.Adding a state calculates the expectation value of the state for each of the supplied
e_ops
and stores the result in.expect
.The state is recorded in
.states
and.final_state
if specified by the supplied result options. Parameters:
 tfloat
The time of the added state.
 statetypically a
Qobj
The state a time
t
. Usually this is aQobj
with suitable dimensions, but it subclasses of result might support other forms of the state. .. note::
The expectation values, i.e.
e_ops
, and states are recorded by the state processors (see.add_processor
).Additional processors may be added by subclasses.
Permutational Invariance
 class Dicke(N, hamiltonian=None, emission=0.0, dephasing=0.0, pumping=0.0, collective_emission=0.0, collective_dephasing=0.0, collective_pumping=0.0)[source]
The Dicke class which builds the Lindbladian and Liouvillian matrix.
 Parameters:
 N: int
The number of twolevel systems.
 hamiltonian
qutip.Qobj
A Hamiltonian in the Dicke basis.
The matrix dimensions are (nds, nds), with nds being the number of Dicke states. The Hamiltonian can be built with the operators given by the jspin functions.
 emission: float
Incoherent emission coefficient (also nonradiative emission). default: 0.0
 dephasing: float
Local dephasing coefficient. default: 0.0
 pumping: float
Incoherent pumping coefficient. default: 0.0
 collective_emission: float
Collective (superradiant) emmission coefficient. default: 0.0
 collective_pumping: float
Collective pumping coefficient. default: 0.0
 collective_dephasing: float
Collective dephasing coefficient. default: 0.0
Examples
>>> from qutip.piqs import Dicke, jspin >>> N = 2 >>> jx, jy, jz = jspin(N) >>> jp = jspin(N, "+") >>> jm = jspin(N, "") >>> ensemble = Dicke(N, emission=1.) >>> L = ensemble.liouvillian()
 Attributes:
 N: int
The number of twolevel systems.
 hamiltonian
qutip.Qobj
A Hamiltonian in the Dicke basis.
The matrix dimensions are (nds, nds), with nds being the number of Dicke states. The Hamiltonian can be built with the operators given by the jspin function in the “dicke” basis.
 emission: float
Incoherent emission coefficient (also nonradiative emission). default: 0.0
 dephasing: float
Local dephasing coefficient. default: 0.0
 pumping: float
Incoherent pumping coefficient. default: 0.0
 collective_emission: float
Collective (superradiant) emmission coefficient. default: 0.0
 collective_dephasing: float
Collective dephasing coefficient. default: 0.0
 collective_pumping: float
Collective pumping coefficient. default: 0.0
 nds: int
The number of Dicke states.
 dshape: tuple
The shape of the Hilbert space in the Dicke or uncoupled basis. default: (nds, nds).
 c_ops()[source]
Build collapse operators in the full Hilbert space 2^N.
 Returns:
 c_ops_list: list
The list with the collapse operators in the 2^N Hilbert space.
 coefficient_matrix()[source]
Build coefficient matrix for ODE for a diagonal problem.
 Returns:
 M: ndarray
The matrix M of the coefficients for the ODE dp/dt = Mp. p is the vector of the diagonal matrix elements of the density matrix rho in the Dicke basis.
 lindbladian()[source]
Build the Lindbladian superoperator of the dissipative dynamics.
 Returns:
 lindbladian
qutip.Qobj
The Lindbladian matrix as a qutip.Qobj.
 lindbladian
 liouvillian()[source]
Build the total Liouvillian using the Dicke basis.
 Returns:
 liouv
qutip.Qobj
The Liouvillian matrix for the system.
 liouv
 pisolve(initial_state, tlist)[source]
Solve for diagonal Hamiltonians and initial states faster.
 Parameters:
 initial_state
qutip.Qobj
An initial state specified as a density matrix of qutip.Qbj type.
 tlist: ndarray
A 1D numpy array of list of timesteps to integrate
 initial_state
 Returns:
 result: list
A dictionary of the type qutip.piqs.Result which holds the results of the evolution.
 class Pim(N, emission=0.0, dephasing=0, pumping=0, collective_emission=0, collective_pumping=0, collective_dephasing=0)[source]
The Permutation Invariant Matrix class.
Initialize the class with the parameters for generating a Permutation Invariant matrix which evolves a given diagonal initial state p as:
dp/dt = Mp
 Parameters:
 N: int
The number of twolevel systems.
 emission: float
Incoherent emission coefficient (also nonradiative emission). default: 0.0
 dephasing: float
Local dephasing coefficient. default: 0.0
 pumping: float
Incoherent pumping coefficient. default: 0.0
 collective_emission: float
Collective (superradiant) emmission coefficient. default: 0.0
 collective_pumping: float
Collective pumping coefficient. default: 0.0
 collective_dephasing: float
Collective dephasing coefficient. default: 0.0
 Attributes:
 N: int
The number of twolevel systems.
 emission: float
Incoherent emission coefficient (also nonradiative emission). default: 0.0
 dephasing: float
Local dephasing coefficient. default: 0.0
 pumping: float
Incoherent pumping coefficient. default: 0.0
 collective_emission: float
Collective (superradiant) emmission coefficient. default: 0.0
 collective_dephasing: float
Collective dephasing coefficient. default: 0.0
 collective_pumping: float
Collective pumping coefficient. default: 0.0
 M: dict
A nested dictionary of the structure {row: {col: val}} which holds non zero elements of the matrix M
 calculate_j_m(dicke_row, dicke_col)[source]
Get the value of j and m for the particular Dicke space element.
 Parameters:
 dicke_row, dicke_col: int
The row and column from the Dicke space matrix
 Returns:
 j, m: float
The j and m values.
 calculate_k(dicke_row, dicke_col)[source]
Get k value from the current row and column element in the Dicke space.
 Parameters:
 dicke_row, dicke_col: int
The row and column from the Dicke space matrix.
 Returns
 ——
 k: int
The row index for the matrix M for given Dicke space element.
 coefficient_matrix()[source]
Generate the matrix M governing the dynamics for diagonal cases.
If the initial density matrix and the Hamiltonian is diagonal, the evolution of the system is given by the simple ODE: dp/dt = Mp.
 isdicke(dicke_row, dicke_col)[source]
Check if an element in a matrix is a valid element in the Dicke space. Dicke row: j value index. Dicke column: m value index. The function returns True if the element exists in the Dicke space and False otherwise.
 Parameters:
 dicke_row, dicke_colint
Index of the element in Dicke space which needs to be checked
 tau_valid(dicke_row, dicke_col)[source]
Find the Tau functions which are valid for this value of (dicke_row, dicke_col) given the number of TLS. This calculates the valid tau values and reurns a dictionary specifying the tau function name and the value.
 Parameters:
 dicke_row, dicke_colint
Index of the element in Dicke space which needs to be checked.
 Returns:
 taus: dict
A dictionary of key, val as {tau: value} consisting of the valid taus for this row and column of the Dicke space element.
Distribution functions
 class Distribution(data=None, xvecs=[], xlabels=[])[source]
A class for representation spatial distribution functions.
The Distribution class can be used to prepresent spatial distribution functions of arbitray dimension (although only 1D and 2D distributions are used so far).
It is indented as a base class for specific distribution function, and provide implementation of basic functions that are shared among all Distribution functions, such as visualization, calculating marginal distributions, etc.
 Parameters:
 dataarray_like
Data for the distribution. The dimensions must match the lengths of the coordinate arrays in xvecs.
 xvecslist
List of arrays that spans the space for each coordinate.
 xlabelslist
List of labels for each coordinate.
 marginal(dim=0)[source]
Calculate the marginal distribution function along the dimension dim. Return a new Distribution instance describing this reduced dimensionality distribution.
 Parameters:
 dimint
The dimension (coordinate index) along which to obtain the marginal distribution.
 Returns:
 dDistributions
A new instances of Distribution that describes the marginal distribution.
 project(dim=0)[source]
Calculate the projection (max value) distribution function along the dimension dim. Return a new Distribution instance describing this reduceddimensionality distribution.
 Parameters:
 dimint
The dimension (coordinate index) along which to obtain the projected distribution.
 Returns:
 dDistributions
A new instances of Distribution that describes the projection.
 visualize(fig=None, ax=None, figsize=(8, 6), colorbar=True, cmap=None, style='colormap', show_xlabel=True, show_ylabel=True)[source]
Visualize the data of the distribution in 1D or 2D, depending on the dimensionality of the underlaying distribution.
Parameters:
 figmatplotlib Figure instance
If given, use this figure instance for the visualization,
 axmatplotlib Axes instance
If given, render the visualization using this axis instance.
 figsizetuple
Size of the new Figure instance, if one needs to be created.
 colorbar: Bool
Whether or not the colorbar (in 2D visualization) should be used.
 cmap: matplotlib colormap instance
If given, use this colormap for 2D visualizations.
 stylestring
Type of visualization: ‘colormap’ (default) or ‘surface’.
 Returns:
 fig, axtuple
A tuple of matplotlib figure and axes instances.
 class TwoModeQuadratureCorrelation(state=None, theta1=0.0, theta2=0.0, extent=[[5, 5], [5, 5]], steps=250)[source]
 update(state)[source]
calculate probability distribution for quadrature measurement outcomes given a twomode wavefunction or density matrix