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  • Open Access

A Heuristic for Matrix Product State Simulation of Out-of-Equilibrium Dynamics of Two-Dimensional Quantum Spin Systems

Salvatore Mandrà*, Brayden Ware, Nikita Astrakhantsev, Sergei Isakov, Benjamin Villalonga, Tom Westerhout, and Kostyantyn Kechedzhi

  • *Contact author: smandra@google.com
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PRX Quantum 7, 033048 – Published 8 September, 2026

DOI: https://doi.org/10.1103/qq2m-v44w

Abstract

Out-of-equilibrium dynamics of non-integrable Hamiltonian many-body quantum systems are characterized by highly entangled wave functions. Near-maximal entanglement arises in systems exhibiting thermalization or pre-thermalization, where the system converges to a steady state with a fixed energy density. Classical simulation of the time dependence of such wave functions requires exponential resources. However, typical computations aim to estimate expectation values of local operators and correlation functions to some expected precision. For thermalizing systems at sufficiently high-energy densities, such computations can be done without storing the full wave function by instead simulating the evolution of the local operator, which requires significantly fewer resources. Nonetheless, constructing such resource-efficient classical algorithms remains a challenge for intermediate energy densities, where simulating both the wave function and operator evolution is costly. In this paper, we propose a heuristic approach to accelerate the convergence of matrix product state (MPS) simulations of expectation values, applicable across a broad range of energy densities. We estimate the desired observables by rescaling the MPS results at low bond dimensions with a factor that depends on the fidelity of the MPS wave function. Using this technique, we simulated the dynamics of the two-dimensional transverse-field Ising model (TFIM) on a 7×8 grid with periodic boundary conditions, using a maximum bond dimension of χ=4096 on a single A100 GPU, as well as the dynamics of the two-dimensional XY model on grids of size up to 9×9. We compare our TFIM results to similar simulations on a digital quantum processor [R. Haghshenas et al. Nature (London) 653, 56 (2026)], demonstrating excellent agreement and confirming the predictive power of our method.

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Simulating how quantum systems evolve over time is a central challenge in physics and quantum computing. As the quantum system evolves, quantum entanglement grows rapidly. Standard classical algorithms, such as matrix product state tensor networks, struggle to keep pace because the memory required to store the state of the system grows exponentially with time. This paper introduces a simple yet powerful rescaling technique that evades this bottleneck. The authors discover that while standard low-memory MPS simulations discard significant quantum state information, expectation values of local physical observables can still be accurately recovered. By multiplying low-fidelity outputs by a correction factor derived from the cumulative truncation fidelity, the rescaled quantities converge to exact physical values far faster than unrescaled MPS simulations. The effectiveness of this approach is demonstrated on the examples of two-dimensional transverse-field Ising and XY spin models on grids up to 11×11, representing key paradigms for non-equilibrium quantum dynamics. Operating on a single graphics processing unit, the rescaled algorithm accurately simulated the dynamics of the TFI model on a 7×8 grid, achieving tight agreement with experimental data from a 56-qubit Quantinuum H2 quantum processor. By proving that accurate physical measurements can be extracted even from highly entangled states that exceed classical representation limits, this work dramatically pushes back the boundaries of classical simulation. It establishes a practical, low-cost classical benchmark to verify quantum hardware and evaluate genuine quantum advantage.

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PRX Quantum 7, 033048– Published 8 September, 2026
  • Received 9 December 2025
  • Accepted 4 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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