Neural Operator Quantum State: A Foundation Model for Quantum Dynamics

Abstract

Capturing the dynamics of quantum many-body systems under time-dependent driving protocols is a central challenge for numerical simulations. Existing methods such as tensor networks and time-dependent neural quantum states, however, must be re-run for every protocol. In this work, we introduce the Neural Operator Quantum State (NOQS) as a foundation model for quantum dynamics. Rather than solving the Schrödinger equation for individual trajectories, our approach aims to \emph{learn the solution operator} that maps entire driving protocols to time-evolved quantum states. Once trained, the NOQS predicts time evolution under unseen protocols in a single forward pass, requiring no additional optimization. We validate NOQS on the two-dimensional Ising model with time-dependent longitudinal and transverse fields, demonstrating accurate prediction not only for unseen in-distribution protocols, but also for qualitatively different, out-of-distribution functional forms of driving. Further, a single NOQS model can be transferred between different temporal resolutions, and can be efficiently fine-tuned with sparse experimental measurements to improve predictions across all observables at negligible cost. Our work introduces a new paradigm for quantum dynamics simulation and provides a practical computational-experimental interface for driven quantum systems.

Figures (6)

• Figure 1: (a) Illustration of the architecture for transformer-based ansatz for quantum state. The input is a spin configuration, $\bm{\sigma} = \{ \sigma_1, \sigma_2, \dots, \sigma_N\}$, $\sigma_i \in \{-1, +1\}$. The physical spins are first mapped to a $d_e$-dimensional latent space via an embedding operation, augmented by positional encodings that contain information about their spatial location. The latent space representations pass through $L_T$ decoder layers. The final latent state is projected by the unembedding layer to yield the log-amplitude $\log(p(\bm{\sigma}; t))$ and phase $\phi(\bm{\sigma}; t)$. (b) Internal structure of each decoder block, showing the masked multi-head self-attention mechanism, residual connections, feed-forward network, and importantly the cross-attention mechanism that attends to temporal information. (c) Time-dependent driving protocols $H(t)$ are processed by the Fourier Neural Operator (FNO) and projected to raw context tokens $\widetilde{M}(t)$, which are then offset by their initial values (Eq. \ref{['eq:rawtocontextvec']}) to respect the initial condition. The transformer wavefunction ansatz attends to the processed context tokens $M(t)$ through a cross-attention mechanism.
• Figure 2: Transverse field $h_x(t)$ and expectation values of local observables for system size $4 \times 4$, benchmarked against exact numerical results. (a) Performance of NOQS on predicting $E(t)$ for in-distribution driving fields unseen during training. The NOQS also generalizes to out-of-distribution driving protocols; for (b) Gaussian pulses and (c) tanh ramps. The NOQS model predicts local observables accurately in all three cases, indicating the learning of time-evolved quantum state under a functional space of driving fields.
• Figure 3: Performance of the NOQS on a $4 \times 8$ lattice ($N=32$), benchmarked against tDMRG ($\chi = 256$). The three driving protocols are the same as those in Fig. \ref{['fig:4by4']}. (a) Energy $E(t)$ obtained from the NOQS predictions, matching almost perfectly with tDMRG results. (b) Average transverse magnetization $\left< X(t) \right>$ for the Gaussian pulse driving protocol. (c)$\langle ZZ(t)\rangle$ for a tanh ramp protocol. Both Gaussian and tanh protocols are out of the training distribution. Despite the exponentially larger Hilbert space compared to a $4 \times 4$ system, the NOQS predictions of local observables and correlators remain accurate.
• Figure 4: Performance of NOQS after fine-tuning, for (a) the Gaussian pulse and (b) tanh driving protocols, at a system size $4 \times 8$. Using only measurements of $X$ and $ZZ$ at four points in time, the Post Fine-Tune predictions become even more accurate for the out-of-distribution fields across the entire time interval.
• Figure 5: Temporal super-resolution of the NOQS on a $4 \times 4$ lattice. The model is trained on $N_t = 200$ time points and evaluated on $N_t = 400$ points, without retraining. (a) Absolute error $|\Delta X(t)| = |\langle X(t)\rangle_{\mathrm{NOQS}} - \langle X(t)\rangle_{\mathrm{exact}}|$ for a Gaussian pulse driving protocol. (b) Absolute error $|\Delta ZZ(t)|$ for a tanh ramp protocol. The smooth error profiles confirm that the NOQS inherits the discretization invariance of the Fourier Neural Operator. Trained on a coarse grid, the NOQS is capable of predicting at finer temporal resolutions accurately.