Gauge Theory plays a crucial role in many areas in science, including high energy physics, condensed matter physics and quantum information science. In quantum simulations of lattice gauge theory, an important step is to construct a wave function that obeys gauge symmetry. In this paper, we have developed gauge equivariant neural network wave function techniques for simulating continuous-variable quantum lattice gauge theories in the Hamiltonian formulation. We have applied the gauge equivariant neural network approach to find the ground state of 2+1-dimensional lattice gauge theory with U(1) gauge group using variational Monte Carlo. We have benchmarked our approach against the state-of-the-art complex Gaussian wave functions, demonstrating improved performance in the strong coupling regime and comparable results in the weak coupling regime.
Local gauge structures play a central role in a wide range of condensed matter systems and synthetic quantum platforms, where they emerge as effective descriptions of strongly correlated phases and engineered dynamics. We introduce a gauge-invariant graph neural network (GNN) architecture for Abelian lattice gauge models, in which symmetry is enforced explicitly through local gauge-invariant inputs, such as Wilson loops, and preserved throughout message passing, eliminating redundant gauge degrees of freedom while retaining expressive power. We benchmark the approach on both
Z2 and
U(1) lattice gauge models, achieving accurate predictions of global observables and spatially resolved quantities despite the nonlocal correlations induced by gauge-matter coupling. We further demonstrate that the learned model serves as an efficient surrogate for semiclassical dynamics in
U(1) quantum link models, enabling stable and scalable time evolution without repeated fermionic diagonalization, while faithfully reproducing both local dynamics and statistical correlations. These results establish gauge-invariant message passing as a compact and physically grounded framework for learning and simulating Abelian lattice gauge systems.
We present a neural network wave function framework for solving non-Abelian lattice gauge theories in a continuous group representation. Using a combination of SU(2) equivariant neural networks alongside an SU(2) invariant, physics-inspired ansatz, we learn a parametrization of the ground state wave function of SU(2) lattice gauge theory in 2+1 and 3+1 dimensions. Our method, performed in the Hamiltonian formulation, has a straightforward generalization to SU(N). We benchmark our approach against a solely invariant ansatz by computing the ground state energy, demonstrating the need for bespoke gauge equivariant transformations. We evaluate the Creutz ratio and average Wilson loop, and obtain results in strong agreement with perturbative expansions. Our method opens up an avenue for studying lattice gauge theories beyond one dimension, with efficient scaling to larger systems, and in a way that avoids both the sign problem and any discretization of the gauge group.
Determining the ground state of non-Abelian lattice gauge theories coupled to dynamical fermions is key to understanding confinement and the phase structure of gauge--matter systems. We present a variational Monte Carlo framework for the ground state of the untruncated fully-continuous SU
(2) lattice gauge theory coupled to dynamical staggered fermions on an
L×L square lattice. We work in the magnetic basis with a neural-network representation of the gauge wavefunction. The fermions are described by a gauge-covariant Gaussian fermionic correction built on a fixed Néel reference state where, for each sampled gauge configuration
U, the correction is generated by a Hermitian operator. This operator is constructed from short Wilson lines and the eigenvectors of the mass--hopping Hamiltonian, with number of variational parameters polynomial in the system size. This Gaussian structure also gives analytical expressions for all fermionic contributions to the energy and related observables in terms of the fermion occupation matrix. The results are validated against strong-coupling perturbation theory, where they recover the expected effective antiferromagnetic spin Hamiltonian. Using this framework, we map a coarse ground state phase diagram in the plane of independent electric and magnetic couplings
(g2,λ) and show that a hysteresis analysis can identify the existence of phase transitions. Restoring the physical relation
λ=4/g2, we characterize how increasing the system size and changing the electric coupling
g2 move the state away from the reference Néel state, for lattice sizes
L=4,6,8. More broadly, the method offers a sign-problem-free variational framework for continuous non-Abelian gauge groups with dynamical matter that should extend to other matter content and higher-dimensional lattices.
This thesis investigates gauge theories by combining theoretical analysis of non-invertible symmetries with computational simulations using neural network quantum states. We first investigate (3+1)d ZN gauge theories with non-invertible duality symmetries that arise from gauging one-form symmetries. These symmetries impose strict constraints on renormalization group flows, often prohibiting symmetry-preserving gapped phases. Our proof demonstrates that a self-dual theory with ZN one-form symmetry must either remain gapless or break self-duality, except when N = k²ℓ with –1 as a quadratic residue modulo ℓ. Lattice gauge theories provide concrete realizations of these constraints. We then focus on the development of neural network quantum states for studying lattice gauge theories. By implementing gauge-invariant neural networks, we compute ground-state wavefunctions for ZN lattice gauge theories in (2+1)d. For Z2 gauge theory, we characterize the continuous confinement transition and identify critical exponents consistent with the Ising universality class. In the Z3 case we observe a weakly first-order transition between the confined and deconfined phases. For both theories, we compute order and disorder parameters and verify the linear confining potential in the confining phase. These results demonstrate that neural network methods provide a powerful tool for studying quantum field theories computationally. By combining these simulations with symmetry-based analysis, this work uncovers new insights into the nonperturbative dynamics and phase structure of gauge theories.
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 networks, including a wide range of architectures such as transformer and recurrent neural network, 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 two-and three-dimensional 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 U(1) lattice gauge theory, obtain the phase diagram for the two-dimensional Z 2 gauge theory, determine the phase transition and the central charge of the 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.