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

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.

NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems

“…Therefore, an ill-conditioned geometric tensor requires many steps of iterative solver, increasing the computational cost. Even then or when using non-iterative methods such as singular value decomposition (SVD), the high condition number can cause instabilities by amplifying noise in the right-hand side of the linear equation [20,21,23]. This is especially true for NQS, which typically feature QGTs with a spectrum spanning many orders of magnitude [60], often making QGT-based algorithms challenging to stabilize [15,21,23].…”

“…To counter that, there is empirical evidence that in some situations, increasing the number of samples used to estimate the QGT and gradients helps to stabilize the solution [23]. Furthermore, it is possible to apply various regularization techniques to the equation.…”

“…They are of particular interest because of their potential to represent highly entangled states in more than one dimension with polynomial resources [9], which is a significant challenge for more established families of variational states. NQS are also flexible: they have been successfully used to determine variational ground states of classical [10] and quantum Hamiltonians [11][12][13][14][15][16][17] as well as excited states [13], to approximate Hamiltonian unitary dynamics [8,[18][19][20][21][22][23], and to solve the Lindblad master equation [24][25][26]. In particular, NQS are currently used in the study of frustrated quantum systems [13,[15][16][17][27][28][29][30], which have so far been challenging to optimize by established numerical techniques.…”

jVMC: Versatile and performant variational Monte Carlo leveraging automated differentiation and GPU acceleration

“…Three possible approaches are outlined below. While either the diagonal shift (Section 2.5.1) or the pseudo-inverse regularization (Section 2.5.2) should typically be used for SR, the effective regularization for real time evolution, e.g., targeted elimination of noisy contributions (Section 2.5.3), is under ongoing investigation and there are no generally established procedures to date [24,51].…”

“…Monte Carlo fluctuations that are blown up by the inversion of small eigenvalues of the Fisher matrix constitute a major source of instabilities when simulating real time evolution [24,51]. A mitigation strategy introduced in Ref.…”