Pseudospectral method for solving PDEs using Matrix Product States

TL;DR

This work develops a pseudospectral approach for time-dependent PDEs by extending Hermite DAFs (HDAF) to matrix product states (MPS), enabling efficient matrix product operator (MPO) representations of derivatives and propagators. It introduces a suite of quantum-inspired time-evolution schemes (explicit/implicit Runge-Kutta, restarted Arnoldi, split-step) and benchmarks them on quantum-quench problems, including harmonic expansion and a double-well potential. The results show that HDAF-MPS can surpass finite-difference methods in accuracy at comparable cost and, due to exponential memory compression, accommodates much larger discretizations than vector-based approaches, with the split-step method delivering the best balance of speed and precision. The findings underline the practicality of quantum-inspired, tensor-network-based solvers for high-dimensional, long-time PDE simulations and point to scalable extensions to more complex systems.

Abstract

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.

Figures (9)

• Figure 1: Errors in the second derivative approximation of a Gaussian function for a varying number of qubits. Dotted lines correspond to the direct implementation of the differentiating operators. Dashed lines implement the procedure specified at the end of Sec. \ref{['sec:hdaf-differentiation']} to limit round-off errors. (a) Finite differences, (b) HDAF.
• Figure 2: Fourier spectrum of $\delta_{M}(x; \sigma)$. Frequencies and widths are normalized to the grid spacing $\Delta x$. The width $\sigma$ is computed for each $M$ according to equation \ref{['eq:sigma-from-M']}.
• Figure 3: One-step evolution for a range of $\Delta t$ and a fixed number of qubits $n=18$. (a) Error $\varepsilon$ (finite difference), (b) Error $\varepsilon$ (HDAF), (c) run time (finite difference), (d) run time (HDAF). The run time is averaged over ten runs.
• Figure 4: Number of qubits $n$ scaling of the split-step one-step evolution for vector-based---HDAF and FFT--- and different tolerance MPS-based HDAF for $\Delta t=0.0001$. (a) Error $\varepsilon$. (b) Run time. (c) Maximum bond dimension $\chi_\text{max}$ . The run time is averaged over ten runs.
• Figure 5: Error $\varepsilon$ scaling with time step $\Delta t$ for the split-step one-step evolution for vector-based---HDAF and FFT--- and different tolerance MPS-based HDAF for $n=20$.