seemps.evolution.euler

seemps.evolution.euler(H: MPO, time: float | tuple[float | int | floating | integer, float | int | floating | integer] | Sequence[float | int | floating | integer] | ndarray[tuple[Any, ...], dtype[_ScalarT]], state: MPS, steps: int = 1000, strategy: Strategy = DEFAULT_STRATEGY, callback: Callable[[float, MPS], Any] | None = None, itime: bool = False)

Solve a Schrodinger equation using the Euler method.

Integrates the Schrodinger equation in real or imaginary time using a first order Euler method. The equation is defined as

\[i\frac{d}{dt}\psi = H(t) \psi\]

in real time and as

\[\frac{d}{dt}\psi = -H(t) \psi\]

in imaginary time evolution.

The integration algorithm is very simple. It is

\[\psi(t_{n+1}) = \psi(t_{n}) - i \delta t H \psi(t_{n})\]

for real time and

\[\psi(t_{n+1}) = \psi(t_{n}) - \delta t H \psi(t_{n})\]

for imaginary time evolution. The integration step is deduced from the arguments as explained below.

The time denotes the integration interval.

• If it is a single number T, the initial condition is \(t=0\) and the evolution proceeds in steps of \(\delta{t}=T/N\) where N=steps.

• If time is a tuple, it contains the initial and final time, and the number of integration steps is deduced from N=steps as \(\delta{t}=T/N\)

• If time is a sequence of numbers, starting with the initial condition, and progressing in time steps time[n+1]-time[n].

The Euler algorithm is a very bad integrator and is offered only for illustrative purposes.

Parameters:

HMPO

Hamiltonian in MPO form.

timeReal | tuple[Real, Real] | Sequence[Real]

Integration interval, or sequence of time steps.

stateMPS

Initial guess of the ground state.

stepsint, default = 1000

Integration steps, if not defined by t_span.

strategyStrategy, default = DEFAULT_STRATEGY

Truncation strategy for MPO and MPS algebra.

callbackCallable[[float, MPS], Any] | None

A callable called after each iteration (defaults to None).

itimebool, default = False

Whether to solve the imaginary time evolution problem.

Returns:

resultMPS | list[Any]

Final state after evolution or values collected by callback