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. 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 x 10 lattices with evolution times beyond T = 10 using modest computational resources. 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. 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.
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. 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 x 10 lattices with evolution times beyond T = 10 using modest computational resources. 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. 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.
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 Z 2 gauge theory; (III) vison confinement dynamics in a Z 2 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.
We propose a novel tensor-network method to achieve accurate and efficient simulations of Abelian lattice gauge theories (LGTs) in (2+1)D for both ground-state and real-time dynamics. The first key is to identify a gauge canonical form of gauge-invariant tensor-network states, which already simplifies existing algorithms for (1+1)D LGTs. The second key is to employ the gauge canonical form of a projected entangled-pair state (PEPS) combining with variational Monte Carlo, enabling efficient computations for (2+1)D LGTs. We demonstrate the versatile capability of this approach for accurate ground-state simulation of pure Z_{2}, Z_{3}, and Z_{4} gauge theory, odd-Z_{2} gauge theories, and Z_{2} gauge theory coupled to hard-core bosons, on square lattices up to 32×32. Furthermore, we demonstrate that it allows for accurate simulations of real-time dynamics up to long time, exemplified by the dynamics of elementary excitations of the deconfined Z_{2} gauge field at 10×10. This is also the first example of simulating the real-time dynamics of PEPS with variational Monte Carlo, whose impact may extend beyond gauge theory. Our Letter establishes gauge-invariant PEPS as a powerful approach for both ground-state and dynamical simulations, opening up a new avenue for nonperturbatively studying (2+1)D LGTs.
Variational minimization of tensor network states enables the exploration of low energy states of lattice gauge theories. However, the exact numerical evaluation of high-dimensional tensor network states remains challenging in general. In [E. Zohar and J. I. Cirac, Phys. Rev. D 97, 034510 (2018)] it was shown how, by combining gauged Gaussian projected entangled pair states with a variational Monte Carlo procedure, it is possible to efficiently compute physical observables. In this paper we demonstrate how this approach can be used to investigate numerically the ground state of a lattice gauge theory. More concretely, we explicitly carry out the variational Monte Carlo procedure based on such contraction methods for a pure gauge Kogut-Susskind Hamiltonian with a Z3 gauge field in two spatial dimensions. This is a first proof of principle to the method, which provides an inherent way to increase the number of variational parameters and can be readily extended to systems with physical fermions.
Gauge theories form the basis of our understanding of modern physics—ranging from the description of quarks and gluons to effective models in condensed matter physics. In the nonperturbative regime, gauge theories are conventionally treated discretely as lattice gauge theories. The resulting systems are evaluated with path-integral based Monte Carlo methods. These methods, however, can suffer from the sign problem and do not allow for a direct evaluation of real-time dynamics. In this work, we present a unified and comprehensive framework for gauged Gaussian projected entangled pair states (PEPS), a variational ansatz based on tensor networks. We review the construction of Hamiltonian lattice gauge theories, explain their similarities to PEPS, and detail the construction of the ansatz state. The estimation of ground states is based on a variational Monte Carlo procedure with the PEPS as an ansatz state. This sign-problem-free ansatz can be efficiently evaluated in any dimension with arbitrary gauge groups, and can include dynamical fermionic matter, suggesting new options for the simulation of nonperturbative regimes of gauge theories, including quantum chromodynamics. Published by the American Physical Society 2024
We introduce a method to simulate open quantum many-body dynamics by combining time-dependent variational Monte Carlo (tVMC) with quantum trajectory techniques. Our approach unravels the Lindblad master equation into an ensemble of stochastic Schrödinger equations for a variational ansatz, avoiding the exponential cost of density matrix evolution. The method is compatible with generic ansatzë, including expressive neural-network wavefunctions. We derive the nonlinear stochastic equations of motion for the variational parameters and employ suitable Stratonovich numerical solvers. To validate our approach, we simulate quenches in the locally dissipative long-range Ising model in a transverse field, accurately capturing non-equilibrium magnetization and spin squeezing dynamics relevant to trapped-ion and Rydberg atom experiments. The framework is computationally efficient, scalable on high-performance computing platforms, and can be readily integrated into existing tVMC implementations. This work enables large-scale simulations of complex, dissipative quantum systems in higher dimensions, with broad implications for quantum technology and fundamental science.
In variational Monte Carlo (VMC) calculations of N-site quantum systems with arbitrary all-to-all two-body interactions, evaluating the local energy generally costs O(N3). We introduce a new framework that reduces this cost to O(N) for tensor network states, capable of scalable and accurate computation of real-time dynamics and ground states. As a result, we obtain accurate simulations of the adiabatic real-time protocol of a 10×10 dipolar XY model realized in a Rydberg simulator [C. Chen et al., Nature 616, 691 (2023)], which was previously beyond the reach of classical simulation. Going beyond quantum experiments, we also directly perform ground state VMC to compare with the adiabatic state preparation. Our work demonstrates tensor network VMC as a powerful classical simulator for long-range quantum platforms such as Rydberg and ion-trap simulators, which are currently in urgent need of scalable classical benchmarking tools. As a separate technical contribution, we resolve the pathology of evolving from product states within of tensor network VMC.