Pseudospectral method for solving PDEs using Matrix Product States (2409.02916v2)

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.

• The paper demonstrates integration of HDAF and MPS to achieve exponential accuracy in simulating time-dependent PDEs compared to traditional finite difference methods.
• Four quantum-inspired time evolution algorithms, including split-step and Arnoldi iteration, are benchmarked for efficiency, cost, and precision.
• The proposed HDAF-MPS method offers significant memory efficiency, paving the way for simulating large quantum systems with reduced computational complexity.

Pseudospectral method for solving PDEs using Matrix Product States

The paper by Jorge Gidi, Paula García-Molina, Luca Tagliacozzo, and Juan José García-Ripoll addresses the application of Matrix Product States (MPS) to solve time-dependent partial differential equations (PDEs), focusing on the time-dependent Schrödinger equation. The work leverages Hermite Distributed Approximating Functionals (HDAF) for highly accurate pseudospectral methods to approximate functions of derivatives within an MPS framework, providing a promising approach to overcome traditional computational challenges in simulating large quantum systems.

Key Contributions

HDAF-MPS Integration

Central to this work is the integration of HDAF into an MPS finite precision algebra. The authors have extended HDAF, originally developed for reconstructing functions as a series of Hermite polynomials weighted by a Gaussian filter, to represent various differential operators within an MPS formalism. The proposed method offers several benefits:

• Accuracy: HDAF provides exponential accuracy for bandwidth-limited functions, significantly surpassing finite difference methods. This accuracy is particularly advantageous for MPS representations, which similarly favor smooth, bandwidth-limited functions.
• Computational Cost: Despite its high precision, the HDAF approach maintains a computational cost comparable to classical finite difference methods. This balance is achieved by tuning the parameters $M$ (highest polynomial order) and $\sigma$ (Gaussian width).

Time Evolution Algorithms

The authors test four types of quantum-inspired algorithms for time evolution:

1. Runge-Kutta Methods: Including explicit methods like Euler and fourth-order Runge-Kutta, and implicit methods like Crank-Nicolson. These are common in PDE solvers, offering varying degrees of accuracy and stability.
2. Arnoldi Iteration: Explicitly restarted Arnoldi iteration approximates the evolution operator within a reduced Krylov subspace, providing a balance between computational cost and accuracy.
3. Split-Step Methods: This method decomposes the Hamiltonian evolution into manageable steps, leveraging the HDAF approximation of the free propagator to avoid Fourier transforms, thus enhancing the efficiency.
4. Vector FFT Methods: The state-of-the-art spectral split-step methods using Fast Fourier Transform (FFT) are benchmarked against MPS implementations.

Benchmarking and Results

The paper uses the expansion of a particle in a quantum quench as the benchmark problem. This scenario involves a rapid increase in space requirements, challenging traditional methods due to exponential scaling in complexity. The main findings include:

• Accuracy and Efficiency: The HDAF-based MPS approach demonstrates higher accuracy compared to finite difference methods at a comparable cost. The HDAF split-step method, in particular, shows excellent balance between cost and accuracy.
• Memory Efficiency: MPS offer an exponential advantage in memory efficiency. This allows for simulations with larger discretizations and expansions compared to traditional vector methods, which are limited by memory constraints.
• Algorithm Comparison: While all tested algorithms improve in precision using HDAF, the split-step method stands out as the best-performing approach due to its subexponential time scaling and efficiency in avoiding Fourier transforms.

Practical Implications and Future Directions

The implications of this research are multifold:

• Quantum Numerical Analysis: The extension of HDAF methods to MPS provides a robust tool for quantum-inspired numerical analysis, particularly for solving time-dependent PDEs. This is crucial for simulating large quantum systems, where traditional methods fail due to exponential scaling.
• Algorithm Development: The proposed methods can be refined and tailored for specific applications wherein the accuracy of differential operators and the efficiency of memory use are critical.
• Higher-Dimensional Problems: Future work could extend these techniques to higher-dimensional PDEs, leveraging the exponential efficiency of MPS representations. This could be particularly beneficial in fields like quantum chemistry, condensed matter physics, and optomechanics.

In summary, the paper provides significant advancements in the numerical solving of PDEs using MPS, specifically through the integration of HDAF for high accuracy and efficiency. The proposed methods not only surpass traditional finite difference approaches but also open new avenues for quantum-inspired simulations on classical computers, potentially bridging towards future applications in computational physics and engineering.

1. Hoffman, D. K., Nayar, N., Sharafeddin, O. A., & Kouri, D. J. (1991). Distributed Approximating Functionals: A New Approach to Accurate Representations of Operator Functions. Journal of Mathematical Physics, 32(4), 1146-1153.
2. Bodmann, D. H., Hoffman, D. K., Kouri, D. J., & Papadakis, K. (2008). Almost-Ideal Low-Pass Filters: Fourier Methods for Hermite Distributed Approximating Functionals. Journal of Computational Physics, 227, 6415-6434.
3. Paeckel, D., Köhler, T. , Swoboda, A. , Manmana, S. R., Schollwöck, U., & Hubig, C. (2019). Time-evolution methods for matrix-product states. Annals of Physics, 411, 167998.
4. Mitra, A. (2018). Quantum Quench Dynamics. Annual Review of Condensed Matter Physics, 9, 245-259.
5. Garcia-Ripoll, J. J., & Cirac, J. I. (2006). Hybrid algorithms for quantum simulation. New Journal of Physics, 8, 269.