Pseudospectral method for solving PDEs using Matrix Product States
<|reference_start|>Pseudospectral method for solving PDEs using Matrix Product States: 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.<|reference_end|>
@article{gidi2024pseudospectral,
    title={Pseudospectral method for solving PDEs using Matrix Product States},
    author={Jorge Gidi, Paula Garc'ia-Molina, Luca Tagliacozzo and Juan Jos'e
  Garc'ia-Ripoll},
    journal={arXiv preprint arXiv:2409.02916},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.02916},
    primaryClass={quant-ph cs.NA math.NA}
}