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

TL;DR

This work addresses solving time-independent PDEs by recasting them as extremal eigenvalue problems for Hamiltonians using MPS/QTT tensor networks. It compares five tensor-network approaches—imaginary-time evolution, gradient descent, power method, restarted Arnoldi, and DMRG—under a finite-precision MPO/MPS framework and benchmarks them on harmonic-oscillator problems in one and two dimensions. The results show exponential memory savings for all MPS-based methods versus traditional vector-based diagonalization, with imaginary-time evolution underperforming relative to gradient-based and Krylov approaches; DMRG and interpolated Arnoldi deliver the best time performance on large problems. The study demonstrates the practical viability of quantum-inspired tensor-network methods for large-scale PDEs and outlines directions for speedups, interpolation strategies, and extensions to other PDEs and many-body problems.

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.

Figures (11)

• Figure 1: (a) Diagrammatic representation of an MPS with open boundary conditions. (b) Scalar product $\braket{\xi | \psi}$ of two states $\ket{\psi}, \ket{\xi}$ in the MPS representation. (c) Contraction of an MPS and an MPO.
• Figure 2: MPO elements for the one-dimensional $\hat{x}$ operator with $\chi=2$. The figure depicts the non-zero tensors' elements of the MPO. Middle tensors (b) are rank-4 tensors, while the first (a) and last (c) tensor have rank-3. If the tensor is equal to a delta function it is represented as a straight line.
• Figure 3: Evolution of the figures of merit versus execution time, when solving the one-dimensional quantum harmonic oscillator PDE over the interval $x\in[-L/2,L/2)$, with $L=10$, using a discretization with $n=8$ qubits and $\Delta x=L/2^n$. We plot (a) the absolute error $\varepsilon$ in the estimation of the eigenvalue, (b) the norm-1 distance, (d) the infidelity \ref{['eq:infidelity']} with respect to the numerically exact solution, and (c) the standard deviation \ref{['eq:std']} of the Hamiltonian over the computed eigenstate.
• Figure 4: Execution time of each algorithm averaged over 10 runs, to estimate the Hamiltonian's lowest eigenvalue with an error below $10^{-10}$, when solving one-dimensional harmonic oscillator over the interval $x\in[-L/2,L/2)$, with $L=10$ and a discretization of $n$ qubits, $\Delta{x}=L/2^n$.
• Figure 5: Results for different truncation tolerances in the MPS-MPS and MPO-MPS operations, for an Arnoldi diagonalization with $n_v=3$ vectors and a discretization of $n=8$ qubits. NE stands for numerically exact, indicating that the truncation tolerance is the machine precision of floating point operations. (a) Error in the estimation of the energy $\varepsilon$, (b) maximum bond dimension $\chi$ of the resulting MPS solution.