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arXiv:2105.01054 · SciPost Physics

Role of stochastic noise and generalization error in the time propagation of neural-network quantum states

Damian Hofmann, Giammarco Fabiani, Johan H. Mentink, Giuseppe Carleo, Michael A. Sentef

Published May 18, 2022• 30 citations
arXivHTMLPDF

Abstract

Neural-network quantum states (NQS) have been shown to be a suitable variational ansatz to simulate out-of-equilibrium dynamics in two-dimensional systems using time-dependent variational Monte Carlo (t-VMC). In particular, stable and accurate time propagation over long time scales has been observed in the square-lattice Heisenberg model using the Restricted Boltzmann machine architecture. However, achieving similar performance in other systems has proven to be more challenging. In this article, we focus on the two-leg Heisenberg ladder driven out of equilibrium by a pulsed excitation as a benchmark system. We demonstrate that unmitigated noise is strongly amplified by the nonlinear equations of motion for the network parameters, which causes numerical instabilities in the time evolution. As a consequence, the achievable accuracy of the simulated dynamics is a result of the interplay between network expressiveness and measures required to remedy these instabilities. We show that stability can be greatly improved by appropriate choice of regularization. This is particularly useful as tuning of the regularization typically imposes no additional computational cost. Inspired by machine learning practice, we propose a validation-set based diagnostic tool to help determining optimal regularization hyperparameters for t-VMC based propagation schemes. For our benchmark, we show that stable and accurate time propagation can be achieved in regimes of sufficiently regularized variational dynamics.

DOI
10.21468/scipostphys.12.5.165

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Neural-network quantum states (NQS) have been shown to be a suitable variational ansatz to simulate out-of-equilibrium dynamics in two-dimensional systems using time-dependent variational Monte Carlo (t-VMC). In particular, stable and accurate time propagation over long time scales has been observed in the square-lattice Heisenberg model using the Restricted Boltzmann machine architecture. However, achieving similar performance in other systems has proven to be more challenging. In this article, we focus on the two-leg Heisenberg ladder driven out of equilibrium by a pulsed excitation as a benchmark system. We demonstrate that unmitigated noise is strongly amplified by the nonlinear equations of motion for the network parameters, which causes numerical instabilities in the time evolution. As a consequence, the achievable accuracy of the simulated dynamics is a result of the interplay between network expressiveness and measures required to remedy these instabilities. We show that stability can be greatly improved by appropriate choice of regularization. This is particularly useful as tuning of the regularization typically imposes no additional computational cost. Inspired by machine learning practice, we propose a validation-set based diagnostic tool to help determining optimal regularization hyperparameters for t-VMC based propagation schemes. For our benchmark, we show that stable and accurate time propagation can be achieved in regimes of sufficiently regularized variational dynamics.
Neural quantum states have recently demonstrated significant potential for simulating quantum dynamics beyond the capabilities of existing variational ansätze. However, studying strongly driven quantum dynamics with neural networks has proven challenging so far. Here, we focus on assessing several sources of numerical instabilities that can appear in the simulation of quantum dynamics based on the time-dependent variational principle (TDVP) with the computationally efficient explicit time integration scheme. Focusing on the restricted Boltzmann machine architecture, we compare solutions obtained by TDVP with analytical solutions and implicit methods as a function of the quench strength. Interestingly, we uncover a quenching strength that leads to a numerical breakdown in the absence of Monte Carlo noise, despite the fact that physical observables don't exhibit irregularities. This breakdown phenomenon appears consistently across several different TDVP formulations, even those that eliminate small eigenvalues of the Fisher matrix or use geometric properties to recast the equation of motion. We provide evidence that the nature of the instability stems from stiffness of the dynamics of the variational parameters, despite the absence of stiffness in the exact quantum dynamics. We conclude that alternative methods need to be developed to leverage the computational efficiency of explicit time integration of the TDVP equations for simulating strongly nonequilibrium quantum dynamics with neural-network quantum states.
We investigate the challenge of classical simulation of unitary quantum dynamics with variational Monte Carlo approaches, addressing the instabilities and high computational demands of existing methods. By systematically analyzing the convergence of stochastic infidelity optimizations, examining the variance properties of key stochastic estimators, and evaluating the error scaling of multiple dynamical discretization schemes, we provide a thorough formalization and significant improvements to the projected time-dependent Variational Monte Carlo (p-tVMC) method. We benchmark our approach on a two-dimensional Ising quench, achieving state-of-the-art performance. This work establishes p-tVMC as a powerful framework for simulating the dynamics of large-scale two-dimensional quantum systems, surpassing alternative VMC strategies on the investigated benchmark problems.
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.
Neural quantum states are a promising framework for simulating many-body quantum dynamics, as they can represent states with volume-law entanglement. As time evolves, the neural network parameters are typically optimized at discrete time steps to approximate the wave function at each point in time. Given the differentiability of the wave function stemming from the Schrödinger equation, here we impose a time-continuous and differentiable parameterization of the neural network by expressing its parameters as linear combinations of temporal basis functions with trainable, time-independent coefficients. We test this ansatz, referred to as the smooth neural quantum state (\textit{s}-NQS) with a loss function defined over an extended time interval, under a sudden quench of a non-integrable many-body quantum spin chain. We demonstrate accurate time evolution using a restricted Boltzmann machine as the instantaneous neural network architecture. We show that the parameterization enables accurate simulations with fewer variational parameters, independent of time-step resolution. Furthermore, the smooth neural quantum state also allows us to initialize and evaluate the wave function at times not included in the training set, both within and beyond the training interval.
Real-time dynamics of interacting electrons lies at the interface between quantum mechanics and non-equilibrium physics, governing the microscopic origin of ultrafast phenomena of molecules and nano-materials. Though neural network variational Monte Carlo has achieved unprecedented accuracy for stationary state calculations, its extension to real-time evolution remains challenging. In this work, we introduce the neural basis time-dependent variational Monte Carlo framework, which achieves stable and highly accurate simulations of electron dynamics. By constraining the time evolution to a compact, customized manifold spanned by the neural basis, we effectively bypass instability issues and achieve long-term stable evolution. Moreover, we demonstrate that this framework yields benchmark-quality accuracy in simulating the laser-driven dipole responses of the hydrogen atom and a stretched hydrogen molecule, and accurately extracts the dynamic polarizabilities of helium and beryllium atoms. Our work reveals the vast potential of neural network wavefunctions for accurately describing real-time electron dynamics and establishes a promising new route for first-principles simulations of complex, time-dependent electronic phenomena.
Quantum many-body problems are some of the most challenging problems in science and are central to demystifying some exotic quantum phenomena, e.g., high-temperature superconductors. The combination of neural networks (NN) for representing quantum states, coupled with the Variational Monte Carlo (VMC) algorithm, has been shown to be a promising method for solving such problems. However, the run-time of this approach scales quadratically with the number of simulated particles, constraining the practically usable NN to - in machine learning terms - minuscule sizes (<10M parameters). Considering the many breakthroughs brought by extreme NN in the +1B parameters scale to other domains, lifting this constraint could significantly expand the set of quantum systems we can accurately simulate on classical computers, both in size and complexity. We propose a NN architecture called Vector-Quantized Neural Quantum States (VQ-NQS) that utilizes vector-quantization techniques to leverage redundancies in the local-energy calculations of the VMC algorithm - the source of the quadratic scaling. In our preliminary experiments, we demonstrate VQ-NQS ability to reproduce the ground state of the 2D Heisenberg model across various system sizes, while reporting a significant reduction of about ×10{\times}10×10 in the number of FLOPs in the local-energy calculation.