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Real-Time Dynamics in Two Dimensions with Tensor Network States via Time-Dependent Variational Monte Carlo Method

Yantao Wu1,* and Jannes Nys2

  • *Contact author: yantaow@iphy.ac.cn
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PRX Quantum 7, 033035 – Published 19 August, 2026

DOI: https://doi.org/10.1103/tggc-8fjx

Abstract

Reliably simulating two-dimensional many-body quantum dynamics with projected entangled pair states (PEPS) has long been a difficult challenge. In this work, we overcome this barrier for low-energy quantum dynamics by developing a stable and efficient time-dependent variational Monte Carlo (tVMC) framework for PEPS. By analytically removing all gauge redundancies of the PEPS manifold and exploiting tensor locality, we obtain a numerically well-conditioned tVMC equation, yielding robust solutions using the efficient Cholesky decomposition. This enables long-time evolution in previously inaccessible regimes. We explain how the difficulties in the traditional approach, particularly those associated with gauge redundancies, are resolved within tVMC. We demonstrate the power and generality of the method through five representative real-time local quench dynamics in two dimensions: (I) chiral edge propagation in a free-fermion Chern insulator; (II) vison propagation in a pure Z2 gauge theory; (III) vison confinement dynamics in a Z2 lattice gauge theory coupled to Higgs field; (IV) fractionalized charge transport in a fractional Chern insulator; and (V) superfluidity and critical velocity in interacting bosons. All simulations are performed on ≥10×10 lattices with evolution times beyond T=10 using modest computational resources. Where exact benchmarks exist (case I), PEPS-tVMC matches free-fermion dynamics with high accuracy up to T=12. In addition, we also simulate the paradigmatic dynamics of the Ising model following a global quench at the critical transverse field, and obtain, with modest bond dimension, agreement with previous results. These results demonstrate PEPS-tVMC as a practical and versatile tool for real-time quantum dynamics in two dimensions. The method significantly extends the reach of classical tensor-network simulations for studying elementary excitations in quantum many-body systems in real time and provides a valuable computational counterpart to emerging quantum simulators. As a by-product in the development, we also present a new form of minSR, which is more stable and offers a new perspective on tVMC.

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Popular Summary

Understanding how quantum many-body systems evolve in real time is a central goal of modern physics. It is important for interpreting experiments in quantum materials, benchmarking quantum simulators, and exploring phenomena such as quasiparticle motion, topological transport, and gauge-theory dynamics. In two spatial dimensions, however, reliable classical simulation is extremely difficult. Exact methods are limited to small systems, while Monte Carlo methods generally suffer from a dynamical sign problem. Tensor network states offer a powerful alternative, but applying projected entangled pair states (PEPS), the natural tensor-network ansatz for two-dimensional systems, to real-time dynamics has remained technically challenging. In this work, we develop a stable time-dependent variational Monte Carlo framework for evolving PEPS in real time. The key difficulty we address is the large gauge redundancy of the PEPS parametrization: many different tensor parameters can represent exactly the same physical quantum state. These redundant directions make the variational equations of motion ill-conditioned and can destabilize time evolution. We show how to identify and remove these gauge directions analytically, and we also explain how the reduced minSR formulation avoids them in large-scale simulations. This leads to a well-conditioned time-evolution problem that can be solved efficiently with Cholesky decomposition. We demonstrate the resulting PEPS–tVMC method on a range of two-dimensional dynamical problems. These include chiral edge propagation in a Chern insulator, vison motion and confinement in Z2 lattice gauge theories, fractional charge pumping in a bosonic fractional Chern insulator, superfluid flow around a moving obstacle, and a benchmark global quench in the transverse-field Ising model. Across these examples, the method reaches lattices of size at least 10×10 and evolution times beyond T=10 with modest computational resources. Our results show that PEPS–tVMC provides a practical classical approach to low-energy real-time dynamics in two-dimensional quantum systems. The method is particularly suited to local quenches and response-type dynamics, where the entanglement growth remains compatible with a PEPS description. It also offers a useful computational counterpart to current and future quantum simulators, where classical benchmarks for two-dimensional nonequilibrium dynamics are urgently needed.

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PRX Quantum 7, 033035– Published 19 August, 2026
  • Received 17 December 2025
  • Revised 15 May 2026
  • Accepted 8 July 2026
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