Instability of explicit time integration for strongly quenched dynamics with neural quantum states

TL;DR

The paper addresses the challenge of simulating strongly driven quantum dynamics with neural quantum states by systematically testing TDVP-based explicit time integration across multiple formulations against exact diagonalization and implicit methods. It uncovers a numerical breakdown at a strong quench (Δ = -2) that occurs even without sampling noise, attributing it to stiffness in the dynamics of variational parameters rather than physical observables. Implicit integration recovers the correct dynamics but at substantial computational cost, and adaptive explicit methods offer limited relief, signaling a need for new stable, scalable approaches. The findings have important implications for designing efficient, reliable nonequilibrium simulations with neural-network quantum states and motivate exploring restricted-subspace or reformulated TDVP strategies. Overall, the work highlights a key bottleneck in explicit TDVP time integration and points toward novel methodological directions for robust NQS dynamics.

Abstract

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.

Figures (8)

• Figure 1: Correlation dynamics of NQS compared to ED, as a function of quenching strength $\Delta$. Red lines indicate NQS results, while the dashed gray lines are obtained with ED. Quench strengths are shown on top of each graph. In almost all cases, NQS results agree very well with ED, except for the breakdown quench of $\Delta=-2$.
• Figure 2: Dynamics of correlation function between neighbouring spins along a quenched bond as a function of time, for two quenches: (a, b) top row $\Delta = 0.5$, and (c, d) bottom row $\Delta=-2$. All full lines are obtained by TDVP time integration of NQS represented by the RBM architecture in Eq. \ref{['eq:rbm']} with $\alpha=1/4$. Dashed gray lines are the ED results. Colors correspond to different formulations: regularization (red), diagonalization (blue), geometric method (green). All NQS results for the well-behaved $\Delta = 0.5$ agree with the ED results. For the breakdown quench $\Delta=-2$, explicit integrator (left) produces wrong results in all formulations, while the implicit integrator (right) produces correct dynamics. The regularization curve in (c) contains a region of interruption from the frozen dynamics, but still does not recover the correct result.
• Figure 3: Correlation dynamics similar to results in Fig. \ref{['fig:breakdown']}, for the same value $\Delta = -2$ of quench strength, but for bigger systems and wider neural networks. On the top graph, results are shown for the $2 \times 2$ lattice, and different values of the network width parameter: $\alpha \in \{ 1,2,3,4,5\}$. The dashed grey line refers to the correct result obtained by ED. The two bottom graphs show the same for $4 \times 4$ and $6 \times 6$ lattices, with $\alpha = 1$. All simulations show a numerical breakdown for this quench, characterized by a loss of dynamics after the first maximum of the correlation function (indicated by the shaded area).
• Figure 4: Spectra of the Quantum Fisher Matrix Eq. \ref{['eq:tdvp']}. The eigenvalues were obtained using three different methods: implicit integration of TDVP using Eq. \ref{['eq:implicit']} and the geometric method formulation Eq. \ref{['eq:tdvp_geo']}, explicit integration of the TDVP using Eq. \ref{['eq:heun']} and the regularization formulation Eq. \ref{['eq:reg_update']}, and fitting the RBM architecture to ED solutions using infidelity optimization. The latter is referred to as exact. Three distinct classes of eigenvalues are observed. Finite eigenvalues (a, b) always have a well-behaved value that doesn't cause a singularity. Vanishing eigenvalues (c) always have a value close to zero in machine precision. On the bottom graph, we also see eigenvalues whose value occasionally becomes small for all methods shown. The inset shows the smallest eigenvalues as a function of integration time step, indicating the dependence of eigenvalue cusp depth as a function of numerical accuracy.
• Figure 5: Results obtained by the adaptive RK45 integrator. (a) Dynamics of the correlation function, showing excellent agreement with the exact results. (b) Dynamics of real (up) and imaginary (down) components of an example variational parameter $w$, showing cusps at the times of correlation maxima. (c) Adaptive time steps as a function of time. Time steps, initially set to $\mathrm{d}t_0 = 10^{-3}$, reduce drastically at the times of correlation maxima, down to a minimal value $\mathrm{d}t_\mathrm{min} = 3.912\cdot 10^{-7}$.