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arXiv · 2409.02916

Pseudospectral method for solving PDEs using Matrix Product States

Jorge Gidi·Paula García-Molina·Luca Tagliacozzo·Juan José García-Ripoll
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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.

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Jorge Gidi, Paula García-Molina, Luca Tagliacozzo, Juan José García-Ripoll. 2024-09-04. Pseudospectral method for solving PDEs using Matrix Product States. https://doi.org/10.1016/j.jcp.2025.114228

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