Quantum Evolution (`quanguru.QuantumToolbox.evolution`)#

Contains functions to create Unitary and open-system super-operators.

Functions#

`Unitary`(Hamiltonian[, timeStep])	Creates Unitary time evolution operator $U(t) := e^{-i\hat{H}t}$ for a given Hamiltonian $\hat{H}$ and time step t.
`Liouvillian`([Hamiltonian, ...])	TODO : I have generalised the functions, docs need to be updated.
`LiouvillianExp`([Hamiltonian, timeStep, ...])	For a time step t, creates Liouvillian $\hat{\mathcal{L}}$ and exponentiate it, or unitary $U(t)$ for a Hamiltonian $\hat{H}$ .
`dissipator`(operatorA[, operatorB, identity, ...])	TODO : I have generalised the functions, docs need to be updated.
`_preSO`(operator[, identity])	Creates pre super-operator $\mathbb{I}\otimes\hat{O}$ for an operator $\hat{O}$ .
`_postSO`(operator[, identity])	Creates pos super-operator $\hat{O}^{T}\otimes\mathbb{I}$ for an operator $\hat{O}$ .
`_prepostSO`(operatorA[, operatorB])	TODO : I have generalised the functions, docs need to be updated.

Function Name	Docstrings	Examples	Unit Tests	Tutorials
Unitary	✅	✅	❌	❌
Liouvillian	✅	✅	❌	❌
LiouvillianExp	✅	✅	❌	❌
dissipator	✅	✅	✅	❌
_preSO	✅	✅	✅	❌
_postSO	✅	✅	✅	❌
_prepostSO	✅	✅	✅	❌

Unitary(Hamiltonian: Union[scipy.sparse._base.spmatrix, numpy.ndarray], timeStep: float = 1.0) → Union[scipy.sparse._base.spmatrix, numpy.ndarray][source]#

Creates Unitary time evolution operator $U(t) := e^{-i\hat{H}t}$ for a given Hamiltonian $\hat{H}$ and time step t.

Keeps sparse/array as sparse/array.

Parameters

Hamiltonian (Matrix) – Hamiltonian of the system
timeStep (float) – time used in the exponentiation (default=1.0)

Returns

Unitary time evolution operator

Return type

Matrix

Examples

>>> Unitary(2*np.pi*sigmaz(), 1).A
array([[1.+2.4492936e-16j, 0.+0.0000000e+00j],
       [0.+0.0000000e+00j, 1.-2.4492936e-16j]])

Liouvillian(Hamiltonian: Optional[Union[scipy.sparse._base.spmatrix, numpy.ndarray]] = None, collapseOperators: Optional[List] = None, decayRates: Optional[List] = None, _double: bool = False) → Union[scipy.sparse._base.spmatrix, numpy.ndarray][source]#

TODO : I have generalised the functions, docs need to be updated. Creates Liouvillian super-operator $\hat{\mathcal{L}} := -i(\hat{H}\otimes\mathbb{I} + \mathbb{I}\otimes\hat{H}) + \sum_{i}\kappa_{i}\hat{\mathcal{D}}(\hat{c}_{i})$ for a Hamiltonian $\hat{H}$ and collapse operators $\{\hat{c}_{i}\}$ (with corresponding decay rates $\{\kappa_{i}\}$ ).

Keeps sparse/array as sparse/array.

Parameters

Hamiltonian (Matrix or None) – Hamiltonian of the system
collapseOperators (list (of Matrix)) – list of collapse operator for Lindblad dissipator terms
decayRates` (list (of float)) – list of decay rates (if not given assumed to be 1)

Returns

Liouvillian super-operator

Return type

Matrix

Examples

>>> Liouvillian(2*np.pi*sigmaz(), [2*np.pi*sigmaz()], [1]).A
array([[  0.         +0.j        ,   0.         +0.j        ,
          0.         +0.j        ,   0.         +0.j        ],
       [  0.         +0.j        , -78.95683521+12.56637061j,
          0.         +0.j        ,   0.         +0.j        ],
       [  0.         +0.j        ,   0.         +0.j        ,
        -78.95683521-12.56637061j,   0.         +0.j        ],
       [  0.         +0.j        ,   0.         +0.j        ,
          0.         +0.j        ,   0.         +0.j        ]])

LiouvillianExp(Hamiltonian: Optional[Union[scipy.sparse._base.spmatrix, numpy.ndarray]] = None, timeStep: float = 1.0, collapseOperators: Optional[List] = None, decayRates: Optional[List] = None, exp: bool = True, _double: bool = False) → Union[scipy.sparse._base.spmatrix, numpy.ndarray][source]#

For a time step t, creates Liouvillian $\hat{\mathcal{L}}$ and exponentiate it, or unitary $U(t)$ for a Hamiltonian $\hat{H}$ .

Keeps sparse/array as sparse/array.

Parameters

Hamiltonian (Matrix or None) – Hamiltonian of the system
timeStep (float) – time used in the exponentiation (default=1)
collapseOperators (list (of Matrix)) – list of collapse operator for Lindblad dissipator terms
decayRates (list (of float)) – list of decay rates (if not given assumed to be 1)
exp (bool) – boolean to exponentiate the Liouvillian or not (=True by default)

Returns

(exponentiated) Liouvillian super-operator

Return type

Matrix

Examples

>>> LiouvillianExp(2*np.pi*sigmaz(), 1, [], []).A
array([[1.+2.4492936e-16j, 0.+0.0000000e+00j],
       [0.+0.0000000e+00j, 1.-2.4492936e-16j]])

>>> LiouvillianExp(2*np.pi*sigmaz(), 1, [2*np.pi*sigmaz()], [1]).A
array([[1.00000000e+00+0.00000000e+00j, 0.00000000e+00+0.00000000e+00j,
        0.00000000e+00+0.00000000e+00j, 0.00000000e+00+0.00000000e+00j],
       [0.00000000e+00+0.00000000e+00j, 5.12250228e-35-2.50930241e-50j,
        0.00000000e+00+0.00000000e+00j, 0.00000000e+00+0.00000000e+00j],
       [0.00000000e+00+0.00000000e+00j, 0.00000000e+00+0.00000000e+00j,
        5.12250228e-35+2.50930241e-50j, 0.00000000e+00+0.00000000e+00j],
       [0.00000000e+00+0.00000000e+00j, 0.00000000e+00+0.00000000e+00j,
        0.00000000e+00+0.00000000e+00j, 1.00000000e+00+0.00000000e+00j]])

dissipator(operatorA: Union[scipy.sparse._base.spmatrix, numpy.ndarray], operatorB: Optional[Union[scipy.sparse._base.spmatrix, numpy.ndarray]] = None, identity: Optional[Union[scipy.sparse._base.spmatrix, numpy.ndarray]] = None, _double: bool = False) → Union[scipy.sparse._base.spmatrix, numpy.ndarray][source]#

TODO : I have generalised the functions, docs need to be updated. Creates the Lindblad dissipator super-operator $\hat{\mathcal{D}}(\hat{c}) := (\hat{c}^{\dagger})^{T}\otimes\hat{c} - 0.5(\mathbb{I}\otimes\hat{c}^{\dagger}\hat{c} + \hat{c}^{\dagger}\hat{c}\otimes\mathbb{I})$ for a collapse operator $\hat{c}$ .

Keeps sparse/array as sparse/array.

Parameters

collapseOperator (Matrix) – a collapse operator
identity (Matrix or None) – identity operator (exist for internal use and optimisation)

Returns

Lindblad dissipator

Return type

Matrix

Examples

>>> dissipator(sigmaz()).A
array([[ 0.,  0.,  0.,  0.],
       [ 0., -2.,  0.,  0.],
       [ 0.,  0., -2.,  0.],
       [ 0.,  0.,  0.,  0.]])

>>> dissipator(sigmam()).A
array([[-1. ,  0. ,  0. ,  0. ],
       [ 0. , -0.5,  0. ,  0. ],
       [ 0. ,  0. , -0.5,  0. ],
       [ 1. ,  0. ,  0. ,  0. ]])

_preSO(operator: Union[scipy.sparse._base.spmatrix, numpy.ndarray], identity: Optional[Union[scipy.sparse._base.spmatrix, numpy.ndarray]] = None) → Union[scipy.sparse._base.spmatrix, numpy.ndarray][source]#

Creates pre super-operator $\mathbb{I}\otimes\hat{O}$ for an operator $\hat{O}$ .

Keeps sparse/array as sparse/array.

Parameters

operator (Matrix) – a collapse operator
identity (Matrix or None) – identity operator (exist for internal use and optimisations)

Returns

pre super-operator

Return type

Matrix

Examples

>>> evolution._preSO(sigmam()).A
array([[0., 0., 0., 0.],
       [1., 0., 0., 0.],
       [0., 0., 0., 0.],
       [0., 0., 1., 0.]])

_postSO(operator: Union[scipy.sparse._base.spmatrix, numpy.ndarray], identity: Optional[Union[scipy.sparse._base.spmatrix, numpy.ndarray]] = None) → Union[scipy.sparse._base.spmatrix, numpy.ndarray][source]#

Creates pos super-operator $\hat{O}^{T}\otimes\mathbb{I}$ for an operator $\hat{O}$ .

Keeps sparse/array as sparse/array.

Parameters

operator (Matrix) – a collapse operator
identity (Matrix or None) – identity operator (exist for internal use and optimisations)

Returns

post super-operator

Return type

Matrix

Examples

>>> evolution._postSO(sigmam()).A
array([[0., 0., 1., 0.],
       [0., 0., 0., 1.],
       [0., 0., 0., 0.],
       [0., 0., 0., 0.]])

_prepostSO(operatorA: Union[scipy.sparse._base.spmatrix, numpy.ndarray], operatorB: Optional[Union[scipy.sparse._base.spmatrix, numpy.ndarray]] = None) → Union[scipy.sparse._base.spmatrix, numpy.ndarray][source]#

TODO : I have generalised the functions, docs need to be updated. Creates pre-pos super-operator $(\hat{B}^{\dagger})^{T}\otimes\hat{A}$ for an operator $\hat{O}$ .

Keeps sparse/array as sparse/array.

Parameters

operatorA (Matrix) – collapse operator A
operatorB (Matrix) – collapse operator B

Returns

pre-post super-operator

Return type

Matrix

Examples

>>> evolution._prepostSO(sigmam()).A
array([[0, 0, 0, 0],
       [0, 0, 0, 0],
       [0, 0, 0, 0],
       [1, 0, 0, 0]], dtype=int64)

evolveOpen(initialState, totalTime, timeStep: float = 1.0, Hamiltonian: Optional[Union[scipy.sparse._base.spmatrix, numpy.ndarray]] = None, collapseOperators: Optional[List] = None, decayRates: Optional[List] = None, calcFunc: Optional[Callable] = None, delStates: Optional[bool] = False, _double: bool = False) → Union[scipy.sparse._base.spmatrix, numpy.ndarray][source]#

steadyState(Hamiltonian: Optional[Union[scipy.sparse._base.spmatrix, numpy.ndarray]] = None, collapseOperators: Optional[List] = None, decayRates: Optional[List] = None, _double: bool = False) → Union[scipy.sparse._base.spmatrix, numpy.ndarray][source]#

Quantum Operators (quanguru.QuantumToolbox.operators)

Basic Functions (quanguru.QuantumToolbox.functions)

Quantum Evolution (quanguru.QuantumToolbox.evolution)#

Functions#

Quantum Evolution (`quanguru.QuantumToolbox.evolution`)#