hqs_quantum_solver.evolution

hqs_quantum_solver.evolution#

Implementations of time-evolution functionality quantum systems.

Functions

evolve_using_krylov(hamiltonian, ...[, ...])

Discrete time-evolution using a Krylov-based method.

evolve_using_rk45(hamiltonian, wavefunction, ...)

Discrete time-evolution using the explicit Runge-Kutta 5(4) method.
evolve_using_krylov(hamiltonian: SimpleOperatorProtocol, wavefunction: np.ndarray, time_step: float, tolerance: float = 1e-8, verbose: int = 0) → NDArray#

Discrete time-evolution using a Krylov-based method.

Given an initial wavefunction, this function computes an approximation to the wavefunction after the system has evolved for a time \(\Delta t\), i.e., it computes an approximation to the solution of the Schrödinger at time \(t + \Delta t\), given the wavefunction at time \(t\).

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

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

in units where \(\hbar = 1\). In this equation, the exponential-times-vector expression is approximated using a Krylov method.

Parameters:

  • hamiltonian (SimpleOperatorProtocol) – A Hamiltonian to evolve with.

  • wavefunction (np.ndarray) – The wavefunction \(\psi\) describing the initial state.

  • time_step (float) – The discrete time step \(\Delta t\).

  • tolerance (float) – the tolerance of the wavefunction.

  • verbose (int) – The verbosity level as given by lattice_functions, 0 <= verbose <= 6.

Returns:

The evolved wavefunction

Return type:

ndarray

evolve_using_rk45(hamiltonian: SimpleOperatorProtocol, wavefunction: np.ndarray, time_step: float, rtol: float = 1e-3, atol: float = 1e-6) → NDArray#

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 time $t_1$, 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:

  • hamiltonian (SimpleOperatorProtocol) – The Hamiltonian \(H\).

  • wavefunction (ndarray) – The wavefunction \(\ket{\psi}\) at time \(t = 0\).

  • time_step (float) – The time \(t_1\) at which to compute the approximation.

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

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

Returns:

An approximation to the solution at \(t_1\).

Return type:

ndarray