Approximately-symmetric neural networks for quantum spin liquids

TL;DR

This work proposes and analyzes a family of approximately symmetric neural networks for quantum spin liquid problems that are parameter efficient, scalable, and significantly outperform existing symmetry-unaware neural network architectures.

Abstract

We propose and analyze a family of approximately-symmetric neural networks for quantum spin liquid problems. These tailored architectures are parameter-efficient, scalable, and significantly outperform existing symmetry-unaware neural network architectures. Utilizing the mixed-field toric code and PXP Rydberg Hamiltonian models, we demonstrate that our approach is competitive with the state-of-the-art tensor network and quantum Monte Carlo methods. Moreover, at the largest system sizes (N = 480 for toric code, N=1584 for Rydberg PXP), our method allows us to explore Hamiltonians with sign problems beyond the reach of both quantum Monte Carlo and finite-size matrix-product states. The network comprises an exactly symmetric block following a non-symmetric block, which we argue learns a transformation of the ground state analogous to quasiadiabatic continuation. Our work paves the way toward investigating quantum spin liquid problems within interpretable neural network architectures.

Figures (12)

• Figure 1: (a) The approximately-symmetric NQS architecture for a mixed-field toric code model. The network computes the ground-state amplitude, $\psi_s = \Omega(\sigma(\chi(s)))$, given an input bit string $s$. The circles (squares) represent edge (plaquette) variables. The convolutional neural networks (CNN) $\chi$ and $\Omega$ consist of between 1-16 layers and 2-16 channels (only 1 layer and 4 channels shown for each) and use normalized $\mathbb{C}$-sigmoid and $\mathbb{C}$-ELU non-linearities, respectively \ref{['sec:SM']}. The non-linear map $\sigma$ imposes invariance on all following layers. For training, $\chi$ is initialized to the identity, while $\Omega$ is randomly initialized. The non-symmetric layers "unfatten" the loop symmetry operators of the quantum spin liquid (purple box) in the spirit of quasi-adiabatic continuation hastingswen. These symmetries are then enforced exactly in the following layers. (b) The convergence of the energy density (blue) and the non-local Bricmont-Frölich-Fredenhagen-Marcu string order parameter (red) as a function of training time (step size times iteration number), in a regime of the mixed-field toric code model which suffers from the sign problem. (c) Phase diagram of the toric code (Eq. (\ref{['eq:tchamiltonian']})) as a function of magnetic field strength, with $h_y=0.2$, imposing a sign problem. The phase-transition locations are extracted from finite-size extrapolation of the string order parameter \ref{['sec:SM']}. The red arrow indicates the approximate-to-exact mapping carried out by the non-symmetric block of the network depicted in panel (a).
• Figure 2: Benchmarking the approximately-symmetric NQS applied to the sign-problem-free toric code with $(h_x,h_y,h_z)=(0.2,0.0,0.2)$. (a) The convergence of the energy density as a function of training time (step size times iteration number) for a $4\times 4$ ($N=24$) lattice. Convolutional neural networks (CNNs) (solid purple) become stuck in local minima, while the approximately-symmetric NQS (solid brown) converges to the exact diagonalization result (dashed black). Different shades correspond to different random initializations and network hyperparameters \ref{['sec:SM']}. (b) At larger system sizes, where exact diagonalization is unavailable, we compare our approximately-symmetric NQS to state-of-the-art DMRG (dashed brown) and QMC (dotted brown) calculations. Due to memory constraints, we were unable to obtain converged DMRG results for the $16\times16$ lattice. (Inset): Zoom-in comparing the NQS, DMRG and QMC energy densities for $L=10$. For further analysis under different perturbations and with different network hyperparameters, see \ref{['sec:SM']}. NQS error bars are shown as shading. QMC uncertainty (not shown) is of order $\sim 10^{-4}$ in units of energy density.
• Figure 3:
• Figure 4: (a) The training scheme used to demonstrate interpretability of the approximately-symmetric network. In Step 1, we find the ground state of the model at the exactly symmetric point $(h_x,h_y,h_z) = (0.2, 0.0, 0.0)$ by training only the symmetric part of the network. In Step 2, we turn to $(h_x,h_y,h_z) = (0.2, 0.0, 0.2)$, fix the symmetric weights within the network, and train the non-symmetric block. (b) Energy convergence curve for this training scheme for a $4\times 4$ ($N=24$) lattice, compared with exact diagonalization (dashed lines).
• Figure 5: (a) Convergence of the energy density $E_{\mathrm{GS}} = \langle H \rangle / N$ for approximately symmetric neural networks (red curves), compared to symmetrized RBMs (blue curves) and exact diagonalization (dashed black) on a system of $N = 30$ Rydberg atoms. Inset: schematic depiction of Rydberg atoms trapped in optical tweezers on a ruby lattice semeghini2021probing. (b) Energy density convergence for larger system sizes $N = 234, 408, 630, 1584$ using approximately symmetric neural quantum states (solid curves), compared to finite-cluster DMRG (dashed black curves) with bond dimension $\chi = 1024$ and $100$ sweeps.