Pseudospectral method for solving PDEs using Matrix Product States
For researchers solving high-dimensional PDEs, this work provides a memory-efficient quantum-inspired approach that overcomes vector limitations, enabling larger discretizations and expansions.
The paper extends Hermite Distributed Approximating Functionals (HDAF) to matrix product states (MPS) for solving the time-dependent Schrödinger equation, achieving comparable accuracy and runtime to FFT-based methods while offering exponential memory savings. The MPS HDAF split-step method successfully simulates particle expansion in a double-well potential.
This research focuses on solving time-dependent partial differential equations (PDEs), in particular the time-dependent Schrödinger 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.