def krylovsolve(
H: Qobj,
psi0: Qobj,
tlist: np.array,
krylov_dim: int,
e_ops=None,
options=None,
progress_bar: bool = None,
sparse: bool = False,
):
"""
Time evolution of state vectors for time independent Hamiltonians.
Evolve the state vector ("psi0") finding an approximation for the time
evolution operator of Hamiltonian ("H") by obtaining the projection of
the time evolution operator on a set of small dimensional Krylov
subspaces (m << dim(H)).
The output is either the state vector or the expectation values of
supplied operators ("e_ops") at arbitrary points at ("tlist").
**Additional options**
Additional options to krylovsolve can be set with the following:
* "store_states": stores states even though expectation values are
requested via the "e_ops" argument.
* "store_final_state": store final state even though expectation values are
requested via the "e_ops" argument.
Parameters
----------
H : :class:`qutip.Qobj`
System Hamiltonian.
psi0 : :class: `qutip.Qobj`
Initial state vector (ket).
tlist : None / *list* / *array*
List of times on which to evolve the initial state. If None, nothing
happens but the code won't break.
krylov_dim: int
Dimension of Krylov approximation subspaces used for the time
evolution approximation.
e_ops : None / list of :class:`qutip.Qobj`
Single operator or list of operators for which to evaluate
expectation values.
options : Options
Instance of ODE solver options, as well as krylov parameters.
atol: controls (approximately) the error desired for the final
solution. (Defaults to 1e-8)
nsteps: maximum number of krylov's internal number of Lanczos
iterations. (Defaults to 10000)
progress_bar : None / BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation.
sparse : bool (default False)
Use np.array to represent system Hamiltonians. If True, scipy sparse
arrays are used instead.
Returns
-------
result: :class:`qutip.solver.Result`
An instance of the class :class:`qutip.solver.Result`, which contains
either an *array* `result.expect` of expectation values for the times
`tlist`, or an *array* `result.states` of state vectors corresponding
to the times `tlist` [if `e_ops` is an empty list].
"""
# check the physics
_check_inputs(H, psi0, krylov_dim)
# check extra inputs
e_ops, e_ops_dict = _check_e_ops(e_ops)
pbar = _check_progress_bar(progress_bar)
# transform inputs type from Qobj to np.ndarray/csr_matrix
if sparse:
_H = H.get_data() # (fast_) csr_matrix
else:
_H = H.full().copy() # np.ndarray
_psi = psi0.full().copy()
_psi = _psi / np.linalg.norm(_psi)
# create internal variable and output containers
if options is None:
options = Options(nsteps=10000)
krylov_results = Result()
krylov_results.solver = "krylovsolve"
# handle particular cases of an empty tlist or single element
n_tlist_steps = len(tlist)
if n_tlist_steps < 1:
return krylov_results
if n_tlist_steps == 1: # if tlist has only one element, return it
krylov_results = particular_tlist_or_happy_breakdown(
tlist, n_tlist_steps, options, psi0, e_ops, krylov_results,
pbar) # this will also raise a warning
return krylov_results
tf = tlist[-1]
t0 = tlist[0]
# optimization step using Lanczos, then reuse it for the first partition
dim_m = krylov_dim
krylov_basis, T_m = lanczos_algorithm(_H,
_psi,
krylov_dim=dim_m,
sparse=sparse)
# check if a happy breakdown occurred
if T_m.shape[0] < krylov_dim + 1:
if T_m.shape[0] == 1:
# this means that the state does not evolve in time, it lies in a
# symmetry of H subspace. Thus, theres no work to be done.
krylov_results = particular_tlist_or_happy_breakdown(
tlist,
n_tlist_steps,
options,
psi0,
e_ops,
krylov_results,
pbar,
happy_breakdown=True,
)
return krylov_results
else:
# no optimization is required, convergence is guaranteed.
delta_t = tf - t0
n_timesteps = 1
else:
# calculate optimal number of internal timesteps.
delta_t = _optimize_lanczos_timestep_size(T_m,
krylov_basis=krylov_basis,
tlist=tlist,
options=options)
n_timesteps = int(ceil((tf - t0) / delta_t))
if n_timesteps >= options.nsteps:
raise Exception(
f"Optimization requires a number {n_timesteps} of lanczos iterations, "
f"which exceeds the defined allowed number {options.nsteps}. This can "
"be increased via the 'Options.nsteps' property.")
partitions = _make_partitions(tlist=tlist, n_timesteps=n_timesteps)
if progress_bar:
pbar.start(len(partitions))
# update parameters regarding e_ops
krylov_results, expt_callback, options, n_expt_op = _e_ops_outputs(
krylov_results, e_ops, n_tlist_steps, options)
# parameters for the lazy iteration evolve tlist
psi_norm = np.linalg.norm(_psi)
last_t = t0
for idx, partition in enumerate(partitions):
evolved_states = _evolve_krylov_tlist(
H=_H,
psi0=_psi,
krylov_dim=dim_m,
tlist=partition,
t0=last_t,
psi_norm=psi_norm,
krylov_basis=krylov_basis,
T_m=T_m,
sparse=sparse,
)
if idx == 0:
krylov_basis = None
T_m = None
t_idx = 0
_psi = evolved_states[-1]
psi_norm = np.linalg.norm(_psi)
last_t = partition[-1]
# apply qobj to each evolved state, remove repeated tail elements
qobj_evolved_states = [
Qobj(state, dims=psi0.dims) for state in evolved_states[1:-1]
]
krylov_results = _expectation_values(
e_ops,
n_expt_op,
expt_callback,
krylov_results,
qobj_evolved_states,
partitions,
idx,
t_idx,
options,
)
t_idx += len(partition[1:-1])
pbar.update(idx)
pbar.finished()
if e_ops_dict:
krylov_results.expect = {
e: krylov_results.expect[n]
for n, e in enumerate(e_ops_dict.keys())
}
return krylov_results
