E. M. Fortes, J. M. Ruffinelli, M. Larocca and D. A. Wisniacki.
The Krylov approximation method provides an efficient approach to perform time-evolutions of quantum states for systems with large-dimensional Hilbert spaces.
We are proud to have received a grant to develop this open-source code from Unitary Fund. We have also received funding from the Faculty of Exact and Natural Sciences at the University of Buenos Aires.
This project has also led us to develop a new error-bound for Krylov's method by interpreting Krylov's subspace as a tight-binding model. You can find the details in our preprint Loschmidt echo approach to Krylov-subspace approximation error estimation.
Dependencies:
qutip
numpy
scipy
PyKrylovSolver is not currently in PyPi. However, you can install it directly from GitHub by doing
pip install git+https://github.com/emilianomfortes/krylovsolver/
Time evolution of state vectors for time-independent Hamiltonians.
Evolve the state vector ("psi0"), using an approximation for the time evolution operator ("U") of Hamiltonian ("H") obtained via the projection of U on a set of small dimensional Krylov subspaces (m<<dim(H)).
The output is either the state vectors or the expectation values of supplied operators ("e_ops"), obtained from evolving "psi0" at each time in a list ("tlist").
Additional options
Additional options are available:
"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 provided, it overrides
t0, tf and dt parameters.
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.
if store_states : bool (default False)
If e_ops is provided, store each state vector corresponding to each time
in tlist.
store_final_state : bool (default False)
If e_ops is provided, store the final state vector of the evolution.
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.
tolerance : :float: (default 1e-7)
Minimum bound value for the final state infidelity with respect to
the exact state.
Returns
---------
result: :class:`qutip.Result`
An instance of the class :class:`qutip.Result`, which contains
either an *array* `result.expect` of expectation values for operators
`e_ops` at times specified by the user input `tlist`, or an *array*
`result.states` of state vectors corresponding to the times in `tlist`.
The documentation website is coming up soon.
from PyKrylovsolver.krylovsolver import krylovsolve
from PyKrylovsolver.hamiltonians import h_ising_transverse
from qutip.qobj import Qobj
import numpy as np
N = 8
dim = 2 ** N
psi = np.random.random(dim) + 1j * np.random.random(dim)
psi = psi / np.linalg.norm(psi)
psi = Qobj(psi)
hx, hz = np.ones(N), 0.5 * np.ones(N)
Jx, Jy, Jz = 0 * np.ones(N), 0 * np.ones(N), np.ones(N)
H = h_ising_transverse(N, hx, hz, Jx, Jy, Jz)
tlist = np.linspace(0, 1, 100)
psi_evolved = krylovsolve(H, psi, tlist=tlist, tolerance=1e-2, krylov_dim=5, progress_bar=False, sparse=True)You are welcome to contribute to the development of the algorithm by forking this repository and sending pull requests or filling bug reports at the issues page page. Any code contribution is wished and will get acknowledged in the documentation.
Please cite the original paper available here if you use our error bound approach for the Krylov approximation in your research.