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arXiv:2303.09430

Comparative study of matrix product state/quantized tensor-train algorithms for solving time-independent partial differential equations

Paula García-Molina, Luca Tagliacozzo, Juan José García‐Ripoll

Published Mar 16, 2023• 1 citation
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Abstract

This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism (QTT). This study focuses on Hamiltonian equations, for which five algorithms are introduced: explicit imaginary-time evolution methods, steepest gradient descent in conventional and optimized forms, a power method, and an explicitly restarted Arnoldi method. The first five methods are engineered using a framework of limited-precision linear algebra, in which operators -- i.e., the equation itself -- and vectors are represented using matrix product operator (MPO) and matrix product state (MPS) formalisms, and where operator-vector multiplication and vector addition are approximated with limited resources. All methods are benchmarked using an exactly solvable PDE for a quantum harmonic oscillator in one and two dimensions over a regular grid with up to $2^{30}$ points and compared with the density matrix renormalization group (DMRG) method. Our study reveals that all MPS-based techniques exponentially outperform exact diagonalization techniques based on vectors regarding memory usage. Imaginary-time algorithms are shown to underperform any gradient descent in terms of calibration needs and costs. Finally, MPS DMRG and interpolated Arnoldi-like asymptotically outperform all other methods, including state-of-the-art vector-based exact diagonalization, with significant advantages in time and memory use.

DOI
10.48550/arxiv.2303.09430

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This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism (QTT). This study focuses on Hamiltonian equations, for which five algorithms are introduced: explicit imaginary-time evolution methods, steepest gradient descent in conventional and optimized forms, a power method, and an explicitly restarted Arnoldi method. The first five methods are engineered using a framework of limited-precision linear algebra, in which operators -- i.e., the equation itself -- and vectors are represented using matrix product operator (MPO) and matrix product state (MPS) formalisms, and where operator-vector multiplication and vector addition are approximated with limited resources. All methods are benchmarked using an exactly solvable PDE for a quantum harmonic oscillator in one and two dimensions over a regular grid with up to 2302^{30}230 points and compared with the density matrix renormalization group (DMRG) method. Our study reveals that all MPS-based techniques exponentially outperform exact diagonalization techniques based on vectors regarding memory usage. Imaginary-time algorithms are shown to underperform any gradient descent in terms of calibration needs and costs. Finally, MPS DMRG and interpolated Arnoldi-like asymptotically outperform all other methods, including state-of-the-art vector-based exact diagonalization, with significant advantages in time and memory use.
This work discusses the solution of partial differential equations (PDEs) using matrix product states (MPS). The study focuses on the search for the lowest eigenstates of a Hamiltonian equation, for which five algorithms are introduced: imaginary-time evolution, steepest gradient descent, an improved gradient descent, an implicitly restarted Arnoldi method, and density matrix renormalization group (DMRG) optimization. The first four methods are engineered using a framework of limited-precision linear algebra, where operations between MPS and matrix product operators (MPOs) are implemented with finite resources. All methods are benchmarked using the PDE for a quantum harmonic oscillator in up to two dimensions, over a regular grid with up to 2^{28} points. Our study reveals that all MPSbased techniques outperform exact diagonalization techniques based on vectors, with respect to memory usage. Imaginary-time algorithms are shown to underperform any type of gradient descent, both in terms of calibration needs and costs. Finally, Arnoldilike methods and DMRG asymptotically outperform all other methods, including exact diagonalization, as problem size increases, with an exponential advantage in memory and time usage.
Classical numerical techniques for solving partial differential equations (PDEs) become computationally expensive as the dimension of the discretized differential operator increases. For PDEs giving rise to Sturm--Liouville problems, tensor network (TN) methods can be highly productive: an operator of dimension N×NN\times NN×N can be represented as a matrix product operator (MPO) using only n=log⁡2(N)n=\log_2(N)n=log2​(N) qubits, enabling computation of eigenvalues and eigenvectors via imaginary time evolution (ITE). However, this remains computationally challenging. First, most methods for generating MPOs of large operators without explicit tensor-product structure require prohibitively large memory. Second, the number of Trotterization steps for convergence in conventional ITE increases rapidly with nnn. We present techniques to mitigate both challenges for certain sparse, structured differential operators. To address the first, we construct the MPO by expanding the operator in the Pauli-string basis, enabled by an analytical expression for the Pauli basis coefficients that reduces the memory requirement from O(2n+1)\mathcal{O}(2^{n+1})O(2n+1) to O(2n)\mathcal{O}(2n)O(2n). To address the second, we propose a multistage state-refinement heuristic that accelerates ITE convergence, reducing convergence time by up to two orders of magnitude. Using this TN framework, we compute the first 32 eigenstates of a Laplacian of dimension exceeding 10610^6106 with fidelity above 0.950.950.95 using a 20-qubit MPO. We further validate the method on the 2D anharmonic oscillator and investigate disordered systems, where increasing random potential strength degrades accuracy and limits the approach.
Accurately solving high-dimensional partial differential equations (PDEs) remains a central challenge in computational mathematics. Traditional numerical methods, while effective in low-dimensional settings or on coarse grids, often struggle to deliver the precision required in practical applications. Recent machine learning-based approaches offer flexibility but frequently fall short in terms of accuracy and reliability, particularly in industrial contexts. In this work, we explore a quantum-inspired method based on quantized tensor trains (QTT), enabling efficient and accurate solutions to PDEs in a variety of challenging scenarios. Through several representative examples, we show that the QTT approach can achieve logarithmic scaling in memory and computational cost for linear PDEs when the relevant QTT ranks remain moderate. We also develop QTT space-time formulations that treat time as an additional dimension, allowing the full temporal evolution to be represented and solved globally rather than through sequential time stepping. For the nonlinear Burgers equation, we study both time-stepping and a frozen-coefficient space-time Picard scheme in QTT form, and report empirical convergence behavior on smooth one-dimensional viscous test problems. Additionally, we present a proof-of-concept data-driven workflow within the quantum-inspired framework, in which sampled source data are interpolated into QTT form and then incorporated directly into the structured PDE solver.
This is a set of lectures on tensor networks with a strong emphasis on the core algorithms involving Matrix Product States (MPS) and Matrix Product Operators (MPO). Compared to other presentations, particular care has been given to disentangle aspects of tensor networks from the quantum many-body problem: MPO/MPS algorithms are presented as a way to deal with linear algebra on extremely (exponentially) large matrices and vectors, regardless of any particular application. The lectures include well-known algorithms to find eigenvectors of MPOs (the celebrated DMRG), solve linear problems, and recent learning algorithms that allow one to map a known function into an MPS (the Tensor Cross Interpolation, or TCI, algorithm). The lectures end with a discussion of how to represent functions and perform calculus with tensor networks using the "quantics" representation. They include the detailed analytical construction of important MPOs such as those for differentiation, indefinite integration, convolution, and the quantum Fourier transform. Three concrete applications are discussed in detail: the simulation of a quantum computer (either exactly or with compression), the simulation of a quantum annealer, and techniques to solve partial differential equations (e.g. Poisson, diffusion, or Gross-Pitaevskii) within the "quantics" representation. The lectures have been designed to be accessible to a first-year PhD student and include detailed proofs of all statements.
We investigate quantum-inspired tensor networks (QTNs) for approximating flow maps of hydrodynamic partial differential equations (PDEs). Motivated by the effective low-rank structure that emerges after tensorization of discretized transport and diffusion dynamics, we encode PDE states as matrix product states (MPS) and represent the evolution operator as a structured low-rank matrix product operator (MPO) in tensor-train form (e.g., arising from finite-difference discretizations assembled in MPO form). The MPO is applied directly in MPS form, and rank growth is controlled via canonicalization and SVD-based truncation after each step. We provide theoretical context through standard matrix product properties, including exact MPS representability bounds, local optimality of SVD truncation, and a Lipschitz-type multi-step error propagation estimate. Experiments on one- and two-dimensional linear advection-diffusion and nonlinear viscous Burgers equations demonstrate accurate short-horizon prediction, favorable scaling in smooth diffusive regimes, and error growth in nonlinear multi-step predictions.
Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.