Pseudospectral method for solving PDEs using Matrix Product States

TL;DR

This paper introduces a pseudospectral method using matrix product states (MPS) combined with Hermite Distributed Approximating Functionals (HDAF) to efficiently solve time-dependent PDEs like the Schrödinger equation, offering improved accuracy and memory efficiency.

Contribution

It extends HDAF to MPS, creating a highly accurate pseudospectral approach that surpasses traditional methods in accuracy and memory efficiency for quantum PDE simulations.

Findings

HDAF-MPS outperforms finite difference methods in accuracy.

HDAF avoids Fourier transforms, improving split-step method performance.

MPS provides exponential memory advantage enabling larger discretizations.

Abstract

This research focuses on solving time-dependent partial differential equations (PDEs), in particular the time-dependent Schr\"odinger equation, using matrix product states (MPS). We propose an extension of Hermite Distributed Approximating Functionals (HDAF) to MPS, a highly accurate pseudospectral method for approximating functions of derivatives. Integrating HDAF into an MPS finite precision algebra, we test four types of quantum-inspired algorithms for time evolution: explicit Runge-Kutta methods, Crank-Nicolson method, explicitly restarted Arnoli iteration and split-step. The benchmark problem is the expansion of a particle in a quantum quench, characterized by a rapid increase in space requirements, where HDAF surpasses traditional finite difference methods in accuracy with a comparable cost. Moreover, the efficient HDAF approximation to the free propagator avoids the need for Fourier transforms in split-step methods, significantly enhancing their performance with an improved balance in cost and accuracy. Both approaches exhibit similar error scaling and run times compared to FFT vector methods; however, MPS offer an exponential advantage in memory, overcoming vector limitations to enable larger discretizations and expansions. Finally, the MPS HDAF split-step method successfully reproduces the physical behavior of a particle expansion in a double-well potential, demonstrating viability for actual research scenarios.