Physics-inspired transformer quantum states via latent imaginary-time evolution

TL;DR

This work reframes neural quantum states as latent imaginary-time evolution (LITE) to achieve physically transparent, sign-problem-free ground-state optimization. It identifies that standard Transformer-based NQS induce imaginary-time–dependent effective Hamiltonians, causing overparameterization, and introduces PITQS, which enforces a single static effective Hamiltonian via weight sharing and enhances propagation accuracy with higher-order Trotter–Suzuki decompositions. Demonstrations on the frustrated $J_1$-$J_2$ Heisenberg model and the Hubbard model show that PITQS can match or surpass state-of-the-art TQS accuracy while using substantially fewer variational parameters, highlighting the value of a physics-guided inductive bias. Overall, the framework bridges expressive neural architectures and physically grounded construction, enabling parameter-efficient, transparent design for complex quantum many-body states.

Abstract

Neural quantum states (NQS) are powerful ansätze in the variational Monte Carlo framework, yet their architectures are often treated as black boxes. We propose a physically transparent framework in which NQS are treated as neural approximations to latent imaginary-time evolution. This viewpoint suggests that standard Transformer-based NQS (TQS) architectures correspond to physically unmotivated effective Hamiltonians dependent on imaginary time in a latent space. Building on this interpretation, we introduce physics-inspired transformer quantum states (PITQS), which enforce a static effective Hamiltonian by sharing weights across layers and improve propagation accuracy via Trotter-Suzuki decompositions without increasing the number of variational parameters. For the frustrated $J_1$-$J_2$ Heisenberg model, our ansätze achieve accuracies comparable to or exceeding state-of-the-art TQS while using substantially fewer variational parameters. This study demonstrates that reinterpreting the deep network structure as a latent cooling process enables a more physically grounded, systematic, and compact design, thereby bridging the gap between black-box expressivity and physically transparent construction.

Figures (4)

• Figure 1: PITQS overview. (a) A configuration $n$ is mapped by the encoder $\hat{\mathcal{E}}_\theta$ to initial latent states $z(0)$, evolved by the LITE operator $\hat{\mathcal{U}}_\theta(\beta)$ to $z(\beta)$, and decoded by $\hat{\mathcal{D}}_\theta$ to the wave-function log-amplitude $\log\Psi_\theta(n)$. (b) In PITQS, $\hat{\mathcal{U}}_\theta(\beta)$ is implemented as $L$ weight-shared layers approximating $e^{-\Delta\tau\hat{\mathcal{H}}_\theta}$ with a single effective Hamiltonian $\hat{\mathcal{H}}_\theta$; each step uses a Trotter--Suzuki decomposition (Strang decomposition scheme shown) that alternates the non-local operator $\hat{\mathcal{K}}_\theta$ and the local operator $\hat{\mathcal{V}}_\theta$.
• Figure 2: Optimization of the energy per site (the first 300 iterations are omitted for readability). We compare the PITQS (Strang) ansatz with $N_p=143{,}010$ against a standard TQS with $N_p=155{,}620$.
• Figure 3: Computational costs of TQS and PITQS as a function of total imaginary time $\beta\in\{1.0,2.0,3.0,4.0\}$. (a) Peak GPU memory usage. (b) Computational time per optimization iteration in seconds.
• Figure 4: Energies per site $E^{(\beta)}$ evaluated using $8{,}192$ Monte Carlo samples by truncating the PITQS ansatz using the Strang scheme trained at $\beta=4.0$. Error bars show the Monte Carlo error.