Adaptive quantum dynamics with the time-dependent variational Monte Carlo method

TL;DR

This work tackles the stability and accuracy challenges of simulating quantum dynamics with highly expressive variational wave functions in time-dependent variational Monte Carlo (tVMC). It introduces adaptive tVMC (atVMC), which uses the local-in-time error (LITE) to selectively evolve only the most relevant variational parameters, freezing and unfreezing directions to keep the error below a user-defined threshold. The method demonstrates improved numerical stability and reduced reliance on strong regularization across quantum quenches in the 1D transverse-field Ising model using spin-Jastrow and RBM neural-network states, including large-system tests, by employing collective updates and overparameterization control. The results indicate that atVMC can reliably harness highly expressive ansätze for dynamical simulations, offering a practical path toward stable, accurate quantum dynamics with neural-network quantum states.

Abstract

We introduce an extension of the time-dependent variational Monte Carlo (tVMC) method that adaptively controls the expressivity of the variational quantum state during the simulation of the dynamics. This adaptive tVMC (atVMC) approach is specifically designed to enhance numerical stability when overparameterized variational ansätze lead to ill-conditioned equations of motion. Building on the concept of the local-in-time error (LITE), a measure of the deviation between variational and exact evolution, we introduce a procedure to quantify each parameter's contribution to reducing the LITE, using only quantities already computed in standard tVMC simulations. These relevance estimates guide the selective evolution of only the most significant parameters at each time step, while maintaining a prescribed level of accuracy. We benchmark the algorithm on quantum quenches in the one-dimensional transverse-field Ising model using both spin-Jastrow and restricted Boltzmann machine wave functions, with an emphasis on overparameterized regimes. The adaptive scheme significantly improves numerical stability and reduces the need for strong regularization, enabling reliable simulations with highly expressive variational ansätze.

Figures (5)

• Figure 1: Quench $g = (4 \to 2)$ simulated using atVMC (circular markers, orange color) and tVMC (square markers, blue color) with a spin-Jastrow ansatz on a TFI chain of $32$ spins. (a) Transverse magnetization per spin: variational results compared with the exact solution (dashed line). Error bars are omitted as they are comparable in magnitude to the observable's oscillations. (b) Difference between the variational and the exact transverse magnetization per spin. (b) Number of active variational parameters. (c) Squared LITE during the evolution, compared to the target threshold (horizontal dashed line).
• Figure 2: Quench $g=(4 \to 1.5)$ simulated using atVMC and tVMC with a RBM ansatz with density $d=3$ on a TFI chain of $32$ spins. atVMC simulations are performed with an adaptive time-step scheme, both with (diamond markers, red color) and without (circular markers, orange color) collective parameter updates. (a) Transverse magnetization per spin: variational results compared to the exact solution (dashed line). Error bars are omitted as they are smaller than the marker size. (b) Number of active variational parameters. (c) Squared LITE during the evolution, compared to the target threshold (horizontal dashed line).
• Figure 3: Quench $g=(0.5 \to 1)$ simulated using atVMC with a RBM ansatz with density $d=15$ on a TFI chain of $32$ spins, with collective parameter updates, with (diamond markers, red color) and without (circular markers, orange color) overparameterization control. (a) Energy per spin. (b) Transverse magnetization per spin: atVMC results compared to the exact solution (dashed line). Error bars are omitted as they are comparable in magnitude to the observable's oscillations. (c) Number of active variational parameters selected by the adaptive algorithm. (d) Squared LITE during the evolution, compared to the target threshold (horizontal dashed line).
• Figure 4: Quench $g=(0.5 \to 1)$ simulated using a RBM ansatz with density $d=15$ on a TFI chain of 32 spins. Three approaches are compared: standard tVMC with SNR regularization and an adaptive time-step scheme (square markers, blue color), atVMC with $\lambda^2_\text{LITE} = 10^{-3} \cdot \operatorname{Var}(\hat{H})/\hbar^2$ and $\eta_{\text{sig}}^2 = 5 \cdot 10^{-3}$ (circular markers, orange color), and atVMC with $\lambda^2_\text{LITE} = 10^{-5} \cdot \operatorname{Var}(\hat{H})/\hbar^2$ and $\eta_{\text{sig}}^2 = 10^{-2}$ (diamond markers, red color). (a) Transverse magnetization per spin: variational results compared to the exact solution (dashed line). Error bars are omitted as they are comparable in magnitude to the observable's oscillations. (b) Squared LITE during the evolution, compared to the target thresholds $\lambda^2_\text{LITE} = 10^{-3} \cdot \operatorname{Var}(\hat{H})/\hbar^2$ (horizontal dashed line) and $\lambda^2_\text{LITE} = 10^{-5} \cdot \operatorname{Var}(\hat{H})/\hbar^2$ (horizontal dash-dotted line).
• Figure 5: Quench $g=(4 \to 2)$ simulated using atVMC with a RBM ansatz with density $d=1$ on a TFI chain of $128$ spins. Two approaches are compared: standard tVMC (square markers, blue color) and atVMC with $\lambda^2_\text{LITE} = 10^{-4} \cdot \operatorname{Var}(\hat{H})/\hbar^2$ and $\eta_{\text{sig}}^2 = 10^{-2}$ (circular markers, orange color). (a) Energy per spin. (b) Transverse magnetization per spin: atVMC and tVMC results compared to the exact solution (dashed line). Error bars are omitted as they are comparable in magnitude to the observable's oscillations. (c) Difference between the variational and the exact transverse magnetization per spin for the atVMC simulation and the tVMC simuation. (d) Number of active variational parameters selected by the adaptive algorithm. (e) Squared LITE during the evolution, compared to the target threshold (horizontal dashed line).