While experimental advancements continue to expand the capabilities to control and probe non-equilibrium quantum matter at an unprecedented level, the numerical simulation of the dynamics of correlated quantum systems remains a pivotal challenge - especially in intermediate spatial dimensions. Neural quantum states are emerging as a new computational tool to investigate the time evolution of many-body quantum systems in previously inaccessible regimes. We review the recent progress in the field with a focus on the different time propagation methods, an overview of the reported applications, and a discussion of the major current challenges.
While experimental advancements continue to expand the capabilities to control and probe non-equilibrium quantum matter at an unprecedented level, the numerical simulation of the dynamics of correlated quantum systems remains a pivotal challenge - especially in intermediate spatial dimensions. Neural quantum states are emerging as a new computational tool to investigate the time evolution of many-body quantum systems in previously inaccessible regimes. We review the recent progress in the field with a focus on the different time propagation methods, an overview of the reported applications, and a discussion of the major current challenges.
Neural quantum states provide flexible variational representations of quantum many-body wave functions by combining neural-network parametrizations with Monte Carlo sampling. In this perspective, we review recent advances in their application to condensed-matter systems, focusing on frustrated quantum magnets, interacting lattice fermions, and non-equilibrium dynamics. We discuss the architectures, symmetry constraints, optimization methods, and sampling strategies underlying state-of-the-art calculations, and summarize practical guidelines for reliable simulations. We also examine the principal remaining challenges, including learning non-trivial sign and phase structures, controlling variational bias, enforcing physical symmetries, scaling optimization to large networks, and achieving stable real-time evolution. Finally, we outline promising directions in which neural quantum states may extend the reach of classical simulations of strongly correlated quantum matter.
The efficient numerical simulation of nonequilibrium real-time evolution in isolated quantum matter constitutes a key challenge for current computational methods. This holds in particular in the regime of two spatial dimensions, whose experimental exploration is currently pursued with strong efforts in quantum simulators. In this work we present a versatile and efficient machine learning inspired approach based on a recently introduced artificial neural network encoding of quantum many-body wave functions. We identify and resolve key challenges for the simulation of time evolution, which previously imposed significant limitations on the accurate description of large systems and long-time dynamics. As a concrete example, we study the dynamics of the paradigmatic two-dimensional transverse-field Ising model, as recently also realized experimentally in systems of Rydberg atoms. Calculating the nonequilibrium real-time evolution across a broad range of parameters, we, for instance, observe collapse and revival oscillations of ferromagnetic order and demonstrate that the reached timescales are comparable to or exceed the capabilities of state-of-the-art tensor network methods.
A long-standing goal of nuclear theory is to explain how the structure and dynamics of atomic nuclei and neutron-star matter emerge from the underlying interactions among protons and neutrons. Achieving this goal requires solving the nuclear quantum many-body problem with high accuracy across a wide range of length scales and density regimes. In this review, we discuss how artificial neural network representations of the nuclear many-body wave function have significantly extended the capabilities of continuum quantum Monte Carlo methods. In particular, neural network quantum states enable calculations of larger systems than were previously accessible and provide a flexible framework for capturing phenomena that challenge conventional approaches, including the emergence of nuclear clusters and superfluid phases in dense matter. We highlight recent applications to finite nuclei, infinite nuclear and neutron matter, and dynamical processes relevant to lepton-nucleus and nucleus-nucleus scattering. We also discuss conceptual and methodological connections with condensed matter physics, emphasizing developments in neural network quantum states that bridge strongly correlated systems across disciplines. Together, these developments demonstrate how neural-network methods open new avenues toward unified and accurate descriptions of nuclear structure, matter, and reactions.
Abstract Simulating the dynamics of many-body quantum systems is a significant challenge, especially in higher dimensions where entanglement grows rapidly. Neural quantum states (NQS) offer a promising tool for representing quantum wavefunctions, but their application to time evolution faces scaling challenges. We introduce the time-dependent neural quantum state (t-NQS), a novel approach incorporating explicit time dependence into the neural network ansatz. This framework optimizes a single, time-independent set of parameters to solve the time-dependent Schrödinger equation across an entire time interval. We detail an autoregressive, attention-based transformer architecture and techniques for extending the model’s applicability. To benchmark and demonstrate our method, we simulate quench dynamics in the 2D transverse field Ising model and the time-dependent preparation of the 2D antiferromagnetic state in a Heisenberg model, demonstrating state of the art performance, scalability, and extrapolation to unseen intervals. These results establish t-NQS as a powerful framework for exploring quantum dynamics in strongly correlated systems.
Exponential complexity of many-body wave functions limits accurate numerical simulations of real-time dynamics, especially beyond 1D, where rapid entanglement growth poses severe challenges. Neural Quantum States (NQS) have emerged as a powerful approach for real-time dynamics in 2D, but their scalability and accuracy in 3D have remained an open challenge. Here, we establish NQS as a scalable framework for 3D quantum dynamics by introducing a residual-based convolutional architecture tailored to cubic spin lattices. Focusing on the 3D transverse-field Ising model, we demonstrate that NQS reliably capture distinct quench regimes, including collapse-and-revival dynamics and, most challengingly, the dynamics following a sudden quench to the quantum critical point. We perform finite-rate quenches to the critical point on lattices containing up to 1000 qubits, an unprecedented system size for numerical simulations of real-time dynamics beyond 1D. This enables the first large-scale numerical demonstration of the 3D quantum Kibble--Zurek mechanism. The QKZM in 3D is particularly intriguing because it lies at the upper critical dimension of the Ising universality class, where the standard power laws are modified by logarithmic factors together with prominent sub-leading logarithmic corrections. By deriving these corrections from renormalization-group flow equations up to two-loop order, we obtain a robust data collapse across all simulated system sizes for the correlation function, the excess energy, and the quantum Fisher information, the latter revealing universal multipartite-entanglement dynamics. In all cases, we find compelling agreement with the expected scaling dimensions. Our findings establish NQS as a scalable and reliable tool for exploring nonequilibrium phenomena in 3D quantum matter and for providing numerical benchmarks for 3D quantum simulators.
Reducing computational scaling for simulating non-Markovian dissipative dynamics using artificial neural networks is both a major focus and formidable challenge in open quantum systems. To enable neural quantum states (NQSs), we encode environmental memory in dissipatons (quasiparticles with characteristic lifetimes), yielding the dissipaton-embedded quantum master equation (DQME). The resulting NQS-DQME framework achieves compact representation of many-body correlations and non-Markovian memory. Benchmarking against numerically exact hierarchical equations of motion confirms NQS-DQME maintains comparable accuracy while enhancing scalability and interpretability. This methodology opens new paths to explore non-Markovian open quantum dynamics in previously intractable systems.