Time-dependent Neural Galerkin Method for Quantum Dynamics

TL;DR

The paper tackles the challenge of simulating long-time dynamics in strongly interacting quantum systems by introducing a time-dependent Neural Quantum Galerkin (t-NQG) method based on a global-in-time variational principle. It employs a Galerkin-inspired ansatz $|\Psi_{\theta}(t)\rangle = \sum_{i=0}^M c_i(t)|\phi_i\rangle$ with time-independent neural-network basis states and optimizes a norm- and phase-invariant loss $L_{[0,T]}(\theta)$ that vanishes for exact Schrödinger evolution, yielding a bound $|\epsilon(t)| \le t \sqrt{L_{[0,t]}}$. Matrix elements and overlaps needed for the linear subspace dynamics are estimated via Monte Carlo with a global sampling distribution, enabling efficient long-time simulations and extrapolation beyond the training window. When applied to global quenches in the 1D and 2D Transverse Field Ising model, t-NQG achieves competitive accuracy and reveals ergodicity breaking and potential non-thermalization in 2D, while outperforming some time-stepping variational methods in long-time regimes. This framework opens pathways to study long-time dynamics in strongly correlated quantum systems and can be extended to more expressive neural architectures or noisy quantum-device benchmarks.

Abstract

We introduce a classical computational method for quantum dynamics that relies on a global-in-time variational principle. Unlike conventional time-stepping approaches, our scheme computes the entire state trajectory over a finite time window by minimizing a loss function that enforces the Schrödinger's equation. The variational state is parametrized with a Galerkin-inspired ansatz based on a time-dependent linear combination of time-independent Neural Quantum States. This structure is particularly well-suited for exploring long-time dynamics and enables bounding the error with the exact evolution via the global loss function. We showcase the method by simulating global quantum quenches in the paradigmatic Transverse-Field Ising model in both 1D and 2D, uncovering signatures of ergodicity breaking and absence of thermalization in two dimensions. Overall, our method is competitive compared to state-of-the-art time-dependent variational approaches, while unlocking previously inaccessible dynamical regimes of strongly interacting quantum systems.

Figures (6)

• Figure 1: Sketch of the time-dependent Neural Quantum Galerkin (t-NQG) method for the simulation of quantum dynamics. The approach consists in minimizing the global loss function $L_{[0, T]}(\theta)$ in \ref{['eq:global_loss']} matching $-i H\ket{\Psi_{\theta}(t)}$ (black arrow) and $\ket*{\dot{\Psi}_{\theta}(t)}$ (red arrow) at each time $t \in [0, T]$ in the subspace of the projector $P_{\perp \ket{\Psi_{\theta}(t)}} = 1 - \frac{|\Psi_{\theta}(t)\rangle\langle\Psi_{\theta}(t)|}{\langle\Psi_{\theta}(t)|\Psi_{\theta}(t)\rangle}$. The grey surface represents the variational manifold of the ansatz. The normalizations of the states are not indicated in the figure for simplicity. The ansatz consists of the linear combination of $M+1$ time-independent basis states $\ket{\phi_i}$ parametrized as Neural Quantum States (NQS) with time-dependent coefficients $c_i(t)$.
• Figure 2: Time evolution of the transverse magnetization $\langle \sigma_i^x \rangle (t)$ following global quantum quenches in the TFI model on a $6 \times 6$ (upper panels) and $8 \times 8$ (lower panels) lattices. The system is quenched from the paramagnetically polarized initial state $|\Psi_0\rangle = \bigotimes_{i=1}^{N} |+\rangle_i$ to (a1–b1) the paramagnetic phase at $h = 5$, (a2–b2) the critical point at $h = 3.044 \approx h_c^{\mathrm{2D}}$, and (a3–b3) the ferromagnetic phase at $h = 2$. The basis states are represented by Restricted Boltzmann Machines (RBMs). For the $6 \times 6$ lattice, we employ $M = 18$ basis states and 512 Monte Carlo samples per integration point, while for the $8 \times 8$ lattice we use $M = 8$ basis states and 256 samples. The insets show the evolution of the loss function $\mathcal{L}(t) \equiv \mathcal{L}(|\Psi_{\theta}(t)\rangle)$, normalized with the system size $N$, as a measure of the variational accuracy.
• Figure 3: Relative deviation of the infinite-time transverse magnetization predicted by t-NQG, $\langle \sigma_i^x \rangle_{\infty}$, from its thermal value computed via Quantum Monte Carlo, $\langle \sigma_i^x \rangle_{\text{QMC}}$, for the $6 \times 6$ and $8 \times 8$ lattices across several quenches. The asymptotic exact values are also reported for the $6 \times 6$ system. Error bars on the long-time t-NQG data are assigned by repeating the calculation for 10 independent realizations, while those for the exact results arise from averaging over a finite time window. The inset shows the infinite-time loss function, $\mathcal{L}_{\infty} \equiv \lim_{t \rightarrow \infty} \mathcal{L}(|\Psi_{\theta}(t)\rangle)$, normalized by the system size $N$, which serves as a measure of the variational accuracy.
• Figure 4: Dynamics of the transverse magnetization $\langle \sigma_i^x \rangle(t)$ extrapolated beyond the training interval $[0, T]$ with $T = 2$ (highlighted in light green). The time evolution corresponds to the quench with $h = 3.044 \approx h_c^{\text{2D}}$ in the $6 \times 6$ lattice. We employ $M=6$ RBM basis states, and 512 Monte Carlo samples per integration point.
• Figure 5: Time evolution of the transverse magnetization $\langle \sigma_i^x \rangle (t)$ following global quantum quenches in the TFI model on a 1D spin chain with $N=40$ sites. The system is quenched from the paramagnetically polarized initial state $|\Psi_0\rangle = \bigotimes_{i=1}^{N} |+\rangle_i$ to (a1) the paramagnetic phase at $h = 2$, (a2) the critical point at $h = h_c^{\mathrm{1D}} = 1$, and (a3) the ferromagnetic phase at $h = 1/10$. We employ $M = 20$ RBM basis states and 512 Monte Carlo samples per integration point.