Author: Mathis Beaudoin, 2025
This notebook shows how to compute time propagators with Dysolve using QuTiP. Dysolve is a method to compute time propagators for hamiltonians of the form $H(t) = H_0 + \cos(\omega t)X$ where $H_0$ is some base hamiltonian and $X$ a perturbation. It performs better than other general methods for this class of hamiltonians and, later on, should support more complicated oscillating perturbations. It is still in development with more features to come in the future. For more details on Dysolve, see the corresponding guide in the documentation.
For the moment, Dysolve can be used with the class DysolvePropagator and the function dysolve_propagator from QuTiP's solvers. They follow a similar structure to the class Propagator and the function propagator, another solver that also computes time propagators.
Here, we import the necessary packages.
from qutip.solver.dysolve_propagator import DysolvePropagator, dysolve_propagator
from qutip.solver.propagator import propagator
from qutip import qeye, sigmax, sigmay, sigmaz, tensor, CoreOptions, about
import numpy as np
DysolvePropagator¶We have to define what $H_0$, $X$ and $\omega$ will be. For example, $H(t) = \sigma_z + \cos(10t)\sigma_x$.
H_0 = sigmaz()
X = sigmax()
omega = 10.0
Some options can be defined. max_order will be the order of approximation used when calculating a propagator. The higher this integer is, the more precise the results will be (at a cost of taking more time to calculate). a_tol is simply the absolute tolerance used in the calculations. Finally, a time propagator can be computed using subpropagators of time increment max_dt. If max_dt is set to 0.25, then the propagator $U(1, 0)$ will come from the multiplication of the supropagators $U(0.25, 0)$, $U(0.5, 0.25)$, $U(0.75, 0.5)$ and $U(1, 0.75)$. This allows for more precise results when the evolution is over a long period of time. In our case, we keep a_tol and max_dt to their default value, but we change max_order to 5.
options = {'max_order': 5}
Everything is now defined to initialize an instance.
dy = DysolvePropagator(H_0, X, omega, options=options)
Then, to compute a time propagator, simply call the instance with a given initial time and final time. Also, only a final time can be given and, in that case, the initial time is considered to be 0.
t_i = -1
t_f = 1
U = dy(t_f, t_i)
This returns a single time propagator $U(t_f = 1, t_i = -1)$. To verify that the $U$ is correct, we compare it to what propagator would return.
# Solve using propagator
def X_coeff(t, omega):
return np.cos(omega * t)
H = [H_0, [X, X_coeff]]
args = {'omega': omega}
prop = propagator(
H, [t_i, t_f], args=args, options={"atol": 1e-10, "rtol": 1e-8}
)
# Comparison
with CoreOptions(atol=1e-10, rtol=1e-6):
assert U == prop[1]
dysolve_propagator¶We proceed like the previous example.
# Define the system
H_0 = tensor(sigmax(), sigmaz()) + tensor(qeye(2), sigmay())
X = tensor(qeye(2), sigmaz())
omega = 5.0
# Keep options to default
dysolve_propagator can take more than one time value. If a single time is passed, a single propagator $U(t,0)$ is returned. If a list of times is given, the function will return a list of propagator $[U(\text{times}[i], \text{times}[0])]$ for all $i$.
times = [-0.1, 0, 0.1]
Us = dysolve_propagator(H_0, X, omega, times)
Again, we compare the results with propagator.
# Solve using propagator
def X_coeff(t, omega):
return np.cos(omega * t)
H = [H_0, [X, X_coeff]]
args = {'omega': omega}
props = propagator(
H, times, args=args, options={"atol": 1e-10, "rtol": 1e-8}
)
# Comparison
with CoreOptions(atol=1e-10, rtol=1e-6):
assert Us == props
about()
QuTiP: Quantum Toolbox in Python ================================ Copyright (c) QuTiP team 2011 and later. Current admin team: Alexander Pitchford, Nathan Shammah, Shahnawaz Ahmed, Neill Lambert, Eric Giguère, Boxi Li, Simon Cross, Asier Galicia, Paul Menczel, and Patrick Hopf. Board members: Daniel Burgarth, Robert Johansson, Anton F. Kockum, Franco Nori and Will Zeng. Original developers: R. J. Johansson & P. D. Nation. Previous lead developers: Chris Granade & A. Grimsmo. Currently developed through wide collaboration. See https://github.com/qutip for details. QuTiP Version: 5.2.0.dev0+4e6ded9 Numpy Version: 2.2.5 Scipy Version: 1.15.2 Cython Version: 3.0.12 Matplotlib Version: 3.10.1 Python Version: 3.12.0 Number of CPUs: 4 BLAS Info: Generic INTEL MKL Ext: None Platform Info: Linux (x86_64) Installation path: /home/runner/miniconda3/envs/test-environment-v5/lib/python3.12/site-packages/qutip Installed QuTiP family packages ------------------------------- qutip-qtrl: 0.2.0.dev0+acb71a0 qutip-jax: 0.1.1.dev5 qutip-qip: 0.5.0.dev0+a6d68ca ================================================================================ Please cite QuTiP in your publication. ================================================================================ For your convenience a bibtex reference can be easily generated using `qutip.cite()`