Gauge Invariant and Anyonic Symmetric Transformer and RNN Quantum States for Quantum Lattice Models

TL;DR

This work develops gauge-invariant and anyonic symmetric autoregressive neural networks (AR-NN) to represent quantum lattice states with local constraints, enabling exact sampling and explicit enforcement of gauge or fusion rules. By introducing composite-particle gauge blocks and autoregressive conditioning, the method yields exact representations for ground and excited states of toric codes and X-cube fractons, while providing scalable variational results for models like the U(1) quantum link model, 2D $\mathbb{Z}_N$ gauge theory, and SU(2)$_k$ anyonic chains. The framework supports both ground-state energy optimization and real-time dynamics with variance-reduced losses and transfer learning to handle large systems, producing phase diagrams, central charge estimates, and dynamical observables. These results demonstrate a versatile tool for exploring condensed matter, high-energy physics, and quantum information tasks, with potential extensions to higher spins, 3+1D gauge theories, and topological quantum computation.

Abstract

Symmetries such as gauge invariance and anyonic symmetry play a crucial role in quantum many-body physics. We develop a general approach to constructing gauge invariant or anyonic symmetric autoregressive neural network quantum states, including a wide range of architectures such as Transformer and recurrent neural network (RNN), for quantum lattice models. These networks can be efficiently sampled and explicitly obey gauge symmetries or anyonic constraint. We prove that our methods can provide exact representation for the ground and excited states of the 2D and 3D toric codes, and the X-cube fracton model. We variationally optimize our symmetry incorporated autoregressive neural networks for ground states as well as real-time dynamics for a variety of models. We simulate the dynamics and the ground states of the quantum link model of $\text{U(1)}$ lattice gauge theory, obtain the phase diagram for the 2D $\mathbb{Z}_2$ gauge theory, determine the phase transition and the central charge of the $\text{SU(2)}_3$ anyonic chain, and also compute the ground state energy of the SU(2) invariant Heisenberg spin chain. Our approach provides powerful tools for exploring condensed matter physics, high energy physics and quantum information science.

Figures (33)

• Figure 1: Autoregressive parameterization of wave function with $n$ composite particles. (a) Gauge block. The input $\left\{\widetilde{x}_1, \widetilde{x}_1, \dots, \widetilde{x}_{k-1}\right\}$ is processed through the autoregressive neural network block (see Appendix \ref{['app:nn']} for details), to output amplitude and phase parts. The amplitude part goes through gauge checking, which removes the gauge breaking terms. Afterwards, the square of the amplitude is normalized. (b) Evaluation process. The evaluation process can be performed in parallel for all the input sites. Given the input $\{\widetilde{x}_k\}$, the gauge block simultaneously generates amplitudes and phases for all sites. We then select the correct amplitudes and phases based on the input configuration for each site and construct the wave function from the selected amplitudes and phases. (c) Sampling process. The sampling is done sequentially for each site. We begin with no input and generate the amplitude and phase for the first site. The configuration of the first site is then sampled from the square of the amplitude. Afterwards, we feed the first sample into the gauge block to obtain the second sample. This process continues until we obtain the whole configuration.
• Figure 2: Composite particles for the quantum link model. Each composite particle is defined as $\ket{\sigma_i} \equiv \ket{q_i, e_{i, i+1}}$. We check Gauss's law between $\ket{\sigma_i}$ and $\ket{\sigma_{i+1}}$.
• Figure 3: Variational ground state optimization for the 6-unit-cell (12 sites and 12 links) open-boundary QLM for $m=0$ with and without gauge invariant construction. The gauge invariant autoregressive neural network reaches an accurate ground state while the ansatz without gauge constraints arrives at a non-physical state in the optimization. We use the Transformer neural network with 1 layer, 32 hidden dimensions and the real-imaginary parameterization (see Fig. \ref{['fig:parameterization']}). The neural network is randomly initialized and is trained for 1000 iterations with 12000 samples in each iteration. The neural network architecture and optimization details are discussed in Appendix \ref{['app:nn']}.
• Figure 4: Variational ground state optimization for the open-boundary QLM of different system sizes and different $m$'s with gauge invariant construction. (a) The expectation value of the electric fields averaged over all links, (b) energy and (inset) energy variance per unit cell. We compare our results with the tensor network (TN) results (dashed lines in (a)) from Ref. Rico_2014. The Transformer neural network has 1 layer and 32 hidden dimensions, whereas the RNN has 2 layers and 40 hidden dimensions. For both neural networks, we use the amplitude-phase parameterization (see Fig. \ref{['fig:parameterization']}). The neural networks are randomly initialized. Then they are trained for 3000 iterations with 12000 samples on 40 unit cells. Then, we use transfer learning technique and train the Transformer for 1000 iterations on 80 unit cells and 600 iterations on 120 unit cells. The RNN is then trained for 1000 iterations on 80 and 120 unit cells and 600 iterations on 160 unit cells. The neural network architecture and optimization details are discussed in Appendix \ref{['app:nn']}.
• Figure 5: Dynamics for the 6- and 12-unit-cell (12-24 sites and 12-24 links) open-boundary QLM for $m=0.1$ and $m=2.0$ with and without gauge invariant construction. The dashed curves are the exact results from the exact diagonalization for 6 unit cells. (a) The change in the energy during the dynamics. (b) The expectation value of the electric field averaged over all links. (c) The per step infidelity measure, where $\ket{\Psi}$ and $\ket{\Phi}$ are defined in Sec. \ref{['sec:optimization']}. We use the Transformer neural network with 1 layer, 16 hidden dimensions for 6 unit cells and 32 hidden dimensions for 12 unit cells, and the real-imaginary parameterization (see Fig. \ref{['fig:parameterization']}). The initial state is $\ket{\bullet\rightarrow\circ\rightarrow}$ for each unit cell and we train the neural network using the forward-backward trapezoid method with the time step $\tau = 0.005$, 600 iterations in each time step, and 12000 samples in each iteration. The neural network architecture, initialization and optimization details are discussed in Appendix \ref{['app:nn']}.