Continuous-time parametrization of neural quantum states for quantum dynamics

TL;DR

This work introduces the smooth neural quantum state (s-NQS), a time-continuous variational ansatz in which each network parameter $\vartheta_j(t)$ is a linear combination of smooth temporal basis functions: $\vartheta_j(t)=\sum_{q=0}^{Q-1} g_q(t)\,\theta_{j,q}$. The wavefunction is represented by a restricted Boltzmann machine with complex parameters, and the dynamics are optimized over time intervals using a global loss that aggregates fidelities via the time-evolution operator $\hat{U}(\Delta t)$, which is approximated to second order as $\hat{U}(\Delta t)\approx \mathbb{I}-i\Delta t\hat{H}-\tfrac{1}{2}(\Delta t\hat{H})^2$. The method enables accurate interpolation and extrapolation in time, improves training stability through smooth initializations across successive windows, and demonstrates scalability to lattices up to $L=40$ spins in the tilted Ising model, with results agreeing well with tMPS benchmarks. Potential extensions include higher-order propagation schemes, more expressive architectures, alternative temporal bases, and applications to open or higher-dimensional quantum systems; publicly available code is provided at snqs2025.

Abstract

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.

Figures (5)

• Figure 1: Schematic illustration of the s-NQS approach. Top: the temporal basis functions $\{g_q(t)\}$ are linearly combined with time-independent coefficients $\{\theta_{j,q}\}$ to construct the network parameter $\vartheta_j(t)$. Middle: these parameters determine the variational wave function $|\psi_{\boldsymbol{\vartheta}(t)}\rangle$ (brown dashed line), within a time interval $\tau$ composed of smaller steps $\Delta t$. Bottom: the parameters are optimized by minimizing a global loss function that enforces consistency with the exact evolution state $|\psi(t)\rangle$ (black solid line) over the entire interval, using the time evolution operator $\hat{U}$.
• Figure 2: Quench dynamics of 1D tilted Ising model with open boundary conditions. (a) Time evolution of the transverse magnetization of the middle site $\langle \sigma^x_{L/2} \rangle$; (b) Time evolution of the infidelity between numerical solutions and the exact results. We compare the performance of p-tVMC with $s$-NQS for different values of the temporal expressivity parameter $Q$. Parameters are $\alpha = 5$, $Jdt/\hbar = 0.01$, $L = 10$, $h_x = h_z = 0.3J$ and $J\tau/\hbar= 0.1$.
• Figure 3: Simulation of a quantum quench in the 1D tilted Ising model with $L = 30$ spins and open boundary conditions. (a) Time evolution of the energy density $E/L$ for different time steps $\Delta t$. Inset: effect of successive training rounds at $\Delta t = 10^{-2}$, shown as gradient red curves labeled R1 (light) to R5 (dark). (b) Evolution of the logarithmic loss function $\log(\mathcal{C})$ over training epoch within the first interval. (c) Time evolution of the transverse magnetization $\langle \sigma^x \rangle$, with s-NQS results (curves) compared to tMPS benchmarks (black dots). Red curves correspond to five training rounds at fixed $\Delta t = 10^{-2}$, shown with increasing color intensity; the blue curve shows the final result at reduced $\Delta t = 10^{-3}$. The inset shows the infidelity estimated via Monte Carlo sampling with $50{,}000$ samples, quantifying the overlap between the s-NQS and tMPS wavefunctions. The parameters are $Q = 5, \alpha = 5, J \tau/\hbar = 0.2$, and $JT/\hbar = 2$.
• Figure 4: Time evolution of the transverse magnetization $\langle \sigma^{x} \rangle$ after quench in 1D tilted Ising model with $L = 40$ spins and open boundary conditions. The solid blue line represents the results from s-NQS with $\alpha = 5$ and $Q = 5$, while the black dots correspond to time-dependent matrix product states (tMPS) simulations. The parameters are $J \Delta t/\hbar = 2.5 \times 10^{ - 4}$, $J \tau/\hbar = 0.2$, and $JT/\hbar = 2$.
• Figure 5: System-size dependence of the converged training loss $\log(\mathcal{C})$ in the first time interval, for different choices of time step $\Delta t$. All simulations use the same ansatz parameters ($\alpha = 5$, $Q = 5$) and $5{,}000$ Monte Carlo samples. Each point is averaged over the final $100$ epochs of a $4000$-epoch training run.