hqs_quantum_solver.evolution

hqs_quantum_solver.evolution#

Implementations of time-evolution functionality for quantum systems.

Functions

evolve_krylov(operator, state, t_eval, *[, ...])

Discrete time-evolution using a Krylov-based method.

evolve_rk45(operator, state, t_eval, *[, ...])

Discrete time-evolution using the explicit Runge-Kutta 5(4) method.

evolve_using_krylov(*args, **kwargs)

Discrete time-evolution using a Krylov-based method.

evolve_using_rk45(*args, **kwargs)

Discrete time-evolution using the explicit Runge-Kutta 5(4) method.

Classes

IntegratorResult

Result of a time integration.
class IntegratorResult#

Result of a time integration.

state#

The state vector at the final evaluation time.

Type:

ndarray

observations#

Contains the observation for each value in t_eval.

Type:

list

__init__(state: ndarray, observations: list[T_co]) → None#

Method generated by attrs for class IntegratorResult.

evolve_krylov(operator: SimpleOperatorProtocol, state: np.ndarray, t_eval: float | Sequence[float], *, observer: Observer[T_co] | None = None, progress: Callable[[float], Any] | None = None, tolerance: float = 1e-8, max_krylov_dimension: int = 30, max_time_steps: int = 2**15, additional_orthogonalizations: int = 1, verbose: int = 0) → IntegratorResult[T_co]#

Discrete time-evolution using a Krylov-based method.

Computes an approximation to the solution of the Schrödinger equation,

\[\mathrm{i} \hbar \frac{d}{d t} \ket{\psi(t)} = H \ket{\psi(t)} \,,\]

at times \(t_0, \dots, t_{n-1}\) given the wavefunction at \(t_0\), where we choose the units such that \(\hbar = 1\).

This function uses a Krylov-based method, which means that it computes

\[\psi \mapsto e^{- (\mathrm{i} / \hbar)\, \Delta t\, H} \psi\]

approximatively.

Parameters:

  • operator (SimpleOperatorProtocol) – The linear operator \(H\).

  • state (ndarray) – The wavefunction \(\ket{\psi}\) at time \(t_0\).

  • t_eval (float | Sequence[float]) –

    The values \(t_0, \dots, t_{n-1}\) at which to evaluate the solution. Must be given in ascending order.

    Calling the function with t_eval being just a number, is equivalent to calling the function with [0, t_eval].

  • observer (Observer | None) – Called for every time step \(t_i\). The return values are collected in the observations attribute of the result.

  • progress (Callable[[float], Any] | None) – Called with the current value of t when the time-stepper advances. Can, e.g., be used to display the simulation progress.

  • tolerance (float) – The error tolerance. The method uses a heuristic that should ensure that the error is in the order of the given tolerance.

  • max_krylov_dimension (int) – The maximal number of vectors generated for the Krylov basis.

  • max_time_steps (int) – The maximal number of intermediate time steps to compute.

  • additional_orthogonalizations (int) – The number of additional orthogonalizations to perform. Can be used to improve the orthogonality of the Krylov basis, by setting to a value of one or larger.

  • verbose (int) – The verbosity level as given by lattice_solver, \(0 \le \mathtt{verbose} \le 2\).

Returns:

The final state and observations.

Return type:

IntegratorResult

evolve_rk45(operator: SimpleOperatorProtocol, state: np.ndarray, t_eval: float | Sequence[float], *, observer: Observer[T_co] | None = None, progress: Callable[[float], Any] | None = None, rtol: float = 1e-3, atol: float = 1e-6) → IntegratorResult[T_co]#

Discrete time-evolution using the explicit Runge-Kutta 5(4) method.

Computes an approximation to the solution of the Schrödinger equation,

\[\mathrm{i} \hbar \frac{d}{d t} \ket{\psi(t)} = H \ket{\psi(t)} \,,\]

at times \(t_0, \dots, t_{n-1}\) given the wavefunction at \(t_0\), where we choose the units such that \(\hbar = 1\).

The solver keeps the local error estimate below \(\mathtt{atol}_i + \mathtt{rtol}_i \cdot | \braket{\mathbf{e}_i, \psi} |\).

Parameters:

  • operator (SimpleOperatorProtocol) – The linear operator \(H\).

  • state (ndarray) – The wavefunction \(\ket{\psi}\) at time \(t_0\).

  • t_eval (float | Sequence[float]) –

    The values \(t_0, \dots, t_{n-1}\) at which to evaluate the solution. Must be given in ascending order.

    Calling the function with t_eval being just a number, is equivalent to calling the function with [0, t_eval].

  • observer (Observer | None) – Called for every time step \(t_i\). The return values are collected in the observations attribute of the result.

  • progress (Callable[[float], Any] | None) – Called with the current value of t when the time-stepper advances. Can, e.g., be used to display the simulation progress.

  • rtol (float and array_like) – The relative error tolerance.

  • atol (float and array_like) – The absolute error tolerance.

Returns:

The final state and observations.

Return type:

IntegratorResult

evolve_using_krylov(*args, **kwargs) → ndarray#

Discrete time-evolution using a Krylov-based method.

Compatibility wrapper for evolve_krylov().

Warning

This method is deprected. Use evolve_krylov() instead.

Parameters:
Returns:

The wavefunction after time evolution.

Return type:

ndarray

evolve_using_rk45(*args, **kwargs) → ndarray#

Discrete time-evolution using the explicit Runge-Kutta 5(4) method.

Compatibility wrapper for evolve_rk45().

Warning

This method is deprected. Use evolve_rk45() instead.

Parameters:
Returns:

The wavefunction after time evolution.

Return type:

ndarray

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