Commutator-free quasi-Magnus operators

This repository contains the code to compute the cost of implementing quantum simulation using Commutator-free quasi-Magnus (CFQM) operators, originally introduced in Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems (see pdf here) such as:

$$ \exp(\Omega^{[4]}(h)) = \exp(-i \alpha_1 h H(t_1) - i \alpha_2 h H(t_2))\exp(-i\alpha_2 h H(t_1) -i \alpha_1 h H(t_2)), $$

with $\alpha_1 = \frac{3-2\sqrt{3}}{12}$, $\alpha_2 = \frac{3+2\sqrt{3}}{12}$, $H(t)$ the time-dependent Hamiltonian, and $t_1$ and $t_2$ defined as the roots of the Legendre polynomial of order 2 of the time segment of length $h$.

Navigating the repository

This repository has the following structure

• The cost estimation cost is found in folder CFQMagnus/,
• while coefficients/ contains the coefficients of the CFQMs (see CFQMagnus/compute_coefficients.py for their preparation),
• and results/ saves the error-per-step incurred simulating the CFQM when using time step h.

The main files can are magnus_errors.py , containing the primitives that we need to compute their cost; and main.py, where we compute the error-per-step of each CFQM, which are saved to results/.

The cost minimizaton and associated paper figure generation can be implemented using the figure generation files in CFQMagnus/figure_generation/.

We additionally included a file CFQMagnus/pennylane_simulation.py providing an example of how to implement time simulation of a CFQM in Pennylane, using a Heisenberg model for the sake of an example.

Installation

Navigate to the root directory of this repository and run

pip install -e .

You may also create a new conda environment before this, though the packages we use are standard ones and can be found in the requirements.txt.

How to cite

Find the article in Quantum simulation of time-dependent Hamiltonians via commutator-free quasi-Magnus operators

@misc{casares2024quantum,
      title={Quantum simulation of time-dependent Hamiltonians via commutator-free quasi-Magnus operators},
      author={Pablo Antonio Moreno Casares and Modjtaba Shokrian Zini and Juan Miguel Arrazola},
      year={2024},
      eprint={2403.13889},
      archivePrefix={arXiv},
      primaryClass={quant-ph}
}