hqs_quantum_solver.lindblad#

Implementation of the Lindblad equation.

Classes

class HamiltonianProtocol#

Requirement for an Hamiltonian used in a Linbladian.

__init__(*args, **kwargs)#

property codomain: T_co#

The codomain of the operator.

property domain: T_co#

The domain of the operator.

dot(x: ndarray, out: ndarray | None = None) → ndarray#

Compute the dot product of operator with array; optional out to avoid reallocation.

When represented by NumPy this operation would be A @ x.

Parameters:

• x (np.ndarray) – The vector/array to dot product with.

• out (Optional[np.ndarray]) – Optional array to store results and avoid reallocation.

Returns:

The result of the dot product, A @ x.

Return type:

np.ndarray

dot_add(x: ndarray, out: ndarray, z: float | complex = 1.0) → None#

Compute the dot product of operator with array and add it to out array.

When represented by NumPy this operation would be out += z * (A @ x).

Parameters:

• x (np.ndarray) – The vector/array to dot product with.

• out (np.ndarray) – The array to which the outcome will be added.

• z (complex) – scalar prefactor before addition, defaults to unity.

dot_h(x: ndarray, out: ndarray | None = None) → ndarray#

Compute the dot product of operator adjoint with array, allows optional out.

When represented by NumPy this operation would be A.conj().T @ x. Though transpose returns only a view of a matrix, conjugate returns a copy, hence this method.

Parameters:

• x (np.ndarray) – The vector/array to dot product with.

• out (Optional[np.ndarray]) – Optional array to store results and avoid reallocation.

Returns:

The result of the dot product, A.conj().T @ x.

Return type:

np.ndarray

property dtype: OperatorDType#

The type of the scalar elements of the vector space that is acted on.

rdot(x: ndarray) → ndarray#

Compute the dot product of array with operator.

When represented by NumPy this operation would be x @ A.

Parameters:

x (np.ndarray) – The vector/array to dot product with.

Returns:

The result of the dot product, x @ A.

Return type:

np.ndarray

rdot_h(x: ndarray) → ndarray#

Compute the dot product of array with operator adjoint.

When represented by NumPy this operation would be x @ A.conj().T.

Parameters:

x (np.ndarray) – The vector/array to dot product with.

Returns:

The result of the dot product, x @ A.conj().T.

Return type:

np.ndarray

property shape: Tuple[int, int]#

The shape=(int, int) of the operator.

Returns:

The shape of the operator.

Return type:

Tuple[int, int]

todense() → ndarray#

Return the operator as a dense matrix.

Returns:

The operator as a dense matrix.

Return type:

np.ndarray

class Lindbladian#

The operator $\mathcal{L}$ of the Lindblad equation.

The Lindblad equation is given by

$$\mathrm{i}\hslash \dot{\rho }(t)=\mathcal{L}\rho (t),$$

where

$$\begin{array}{r}\begin{array}{rl}\mathcal{L}\rho =[H,\rho ]& +\mathrm{i}\hslash \sum _{i=0}^{M-1}\sum _{j=0}^{M-1}{\gamma }_{ij}^z(S_i^z\rho S_j^z-\frac{1}{2}{S}_j^z{S}_i^z\rho -\frac{1}{2}\rho S_j^z{S}_i^z)\\ & +\mathrm{i}\hslash \sum _{i=0}^{M-1}\sum _{j=0}^{M-1}{\gamma }_{ij}^{+}(S_i^{+}\rho S_j^{-}-\frac{1}{2}{S}_j^{-}{S}_i^{+}\rho -\frac{1}{2}\rho S_j^{-}{S}_i^{+})\\ & +\mathrm{i}\hslash \sum _{i=0}^{M-1}\sum _{j=0}^{M-1}{\gamma }_{ij}^{-}(S_i^{-}\rho S_j^{+}-\frac{1}{2}{S}_j^{+}{S}_i^{-}\rho -\frac{1}{2}\rho S_j^{+}{S}_i^{-}).\end{array}\end{array}$$

The computation is carried out in units where $\hslash =1$.

__init__(hamiltonian: HamiltonianProtocol, gamma_z: ndarray, gamma_p: ndarray, gamma_m: ndarray) → None#

Lindbladian constructor.

Parameters:

• hamiltonian (SpinOperatorProtocol) – The underlying Hamiltonian $H$.

• gamma_z (ndarray, shape (M,) | ndarray, shape (M,M)) – The ${\gamma }^z$ coefficient matrix. When a vector is given, the argument is interpreted as the diagonal elements of a diagonal matrix.

• gamma_p (ndarray, shape (M,) | ndarray, shape (M,M)) – The ${\gamma }^{+}$ coefficients. When a vector is given, the argument is interpreted as the diagonal elements of a diagonal matrix.

• gamma_m (ndarray, shape (M,) | ndarray, shape (M,M)) – The ${\gamma }^{-}$ coefficients. When a vector is given, the argument is interpreted as the diagonal elements of a diagonal matrix.

apply_von_neumann(rho: ndarray) → ndarray#

Apply operator from the von Neumann equation.

Computes

$$\rho \mapsto [H,\rho ].$$

Parameters:

rho (ndarray, shape (n,n)) – The density matrix to which the operator is applied to.

Returns:

The result of applying the operator to the density matrix rho.

Return type:

ndarray, shape (n, n)

dot(x: ndarray, out: ndarray | None = None) → ndarray#

Implements SimpleOperatorProtocol.dot().

property dtype: OperatorDType#

The type of the scalar elements of the vector space that is acted on.

property hamiltonian: HamiltonianProtocol#

The underlying Hamiltonian $H$.

property shape: tuple[int, int]#

Implements SimpleOperatorProtocol.shape().

class NaiveLindbladian#

The operator $\mathcal{L}$ of the Lindblad equation.

The Lindblad equation is given by

$$\mathrm{i}\hslash \dot{\rho }(t)=\mathcal{L}\rho (t),$$

where

$$\begin{array}{r}\begin{array}{rl}\mathcal{L}\rho =[H,\rho ]& +\mathrm{i}\hslash \sum _{i=0}^{M-1}\sum _{j=0}^{M-1}{\gamma }_{ij}^z(S_i^z\rho S_j^z-\frac{1}{2}{S}_j^z{S}_i^z\rho -\frac{1}{2}\rho S_j^z{S}_i^z)\\ & +\mathrm{i}\hslash \sum _{i=0}^{M-1}\sum _{j=0}^{M-1}{\gamma }_{ij}^{+}(S_i^{+}\rho S_j^{-}-\frac{1}{2}{S}_j^{-}{S}_i^{+}\rho -\frac{1}{2}\rho S_j^{-}{S}_i^{+})\\ & +\mathrm{i}\hslash \sum _{i=0}^{M-1}\sum _{j=0}^{M-1}{\gamma }_{ij}^{-}(S_i^{-}\rho S_j^{+}-\frac{1}{2}{S}_j^{+}{S}_i^{-}\rho -\frac{1}{2}\rho S_j^{+}{S}_i^{-}).\end{array}\end{array}$$

The computation is carried out in units where $\hslash =1$.

This implementation is “naive” in the sense, as it directly evaluates the formula above without any optimization.

property M: int#

The total number of sites.

property Sz: int#

Implements SpinProtocol.Sz().

__init__(hamiltonian: OperatorType, gamma_z: ndarray, gamma_p: ndarray, gamma_m: ndarray) → None#

Lindbladian constructor.

Parameters:

• hamiltonian (SpinOperatorProtocol) – The underlying Hamiltonian $H$.

• gamma_z (ndarray, shape (M,) | ndarray, shape (M,M)) – The ${\gamma }^z$ coefficient matrix. When a vector is given, the argument is interpreted as the diagonal elements of a diagonal matrix.

• gamma_p (ndarray, shape (M,) | ndarray, shape (M,M)) – The ${\gamma }^{+}$ coefficients. When a vector is given, the argument is interpreted as the diagonal elements of a diagonal matrix.

• gamma_m (ndarray, shape (M,) | ndarray, shape (M,M)) – The ${\gamma }^{-}$ coefficients. When a vector is given, the argument is interpreted as the diagonal elements of a diagonal matrix.

apply_von_neumann(rho: ndarray) → ndarray#

Apply operator from the von Neumann equation.

Computes

$$\rho \mapsto [H,\rho ].$$

Parameters:

rho (ndarray, shape (n,n)) – The density matrix to which the operator is applied to.

Returns:

The result of applying the operator to the density matrix rho.

Return type:

ndarray, shape (n, n)

dot(x: ndarray, out: ndarray | None = None) → ndarray#

Implements SimpleOperatorProtocol.dot().

dot_add(x: ndarray, out: ndarray, z: float | complex = 1.0) → None#

Implements SimpleOperatorProtocol.dot_add().

property dtype: OperatorDType#

The type of the scalar elements of the vector space that is acted on.

property hamiltonian: OperatorType#

The underlying Hamiltonian $H$.

property mod_Sz: int#

Implements SpinProtocol.mod_Sz().

property shape: tuple[int, int]#

Implements SimpleOperatorProtocol.shape().

property spin_representation: int#

Implements SpinProtocol.spin_representation().