First-quantized deep neural network techniques are developed for analyzing strongly coupled fermionic systems on the lattice. Using a Slater-Jastrow-inspired ansatz which exploits deep residual networks with convolutional residual blocks, we approximately determine the ground state of spinless fermions on a square lattice with nearest-neighbor interactions. The flexibility of the neural-network ansatz results in a high level of accuracy when compared with exact diagonalization results on small systems, both for energy and correlation functions. On large systems, we obtain accurate estimates of the boundaries between metallic and charge-ordered phases as a function of the interaction strength and the particle density.

Gauge symmetries play a key role in physics appearing in areas such as quantum field theories of the fundamental particles and emergent degrees of freedom in quantum materials. Motivated by the desire to efficiently simulate many-body quantum systems with exact local gauge invariance, gauge equivariant neural-network quantum states are introduced, which exactly satisfy the local Hilbert space constraints necessary for the description of quantum lattice gauge theory with ${\mathbb{Z}}_d$ gauge group and non-Abelian Kitaev $D(G)$ models on different geometries. Focusing on the special case of ${\mathbb{Z}}_2$ gauge group on a periodically identified square lattice, the equivariant architecture is analytically shown to contain the loop-gas solution as a special case. Gauge equivariant neural-network quantum states are used in combination with variational quantum Monte Carlo to obtain compact descriptions of the ground state wave function for the ${\mathbb{Z}}_2$ theory away from the exactly solvable limit, and to demonstrate the confining or deconfining phase transition of the Wilson loop order parameter.

It is possible to couple Dirac-Born-Infeld scalars possessing generalized Galilean internal shift symmetries (Galileons) to nonlinear massive gravity in four dimensions, in such a manner that the interactions maintain the Galilean symmetry. Such a construction is of interest, because it is not possible to couple such fields to massless general relativity in the same way. We show that this theory has the primary constraint necessary to eliminate the Boulware-Deser ghost, thus preserving the attractive properties of both the Galileons and ghost-free massive gravity.

Inspired by the possibility that generative models based on quantum circuits can provide a useful inductive bias for sequence modeling tasks, we propose an efficient training algorithm for a subset of classically simulable quantum circuit models. The gradient-free algorithm, presented as a sequence of exactly solvable effective models, is a modification of the density matrix renormalization group procedure adapted for learning a probability distribution. The conclusion that circuit-based models offer a useful inductive bias for classical datasets is supported by experimental results on the parity learning problem.