Solving the Schrödinger equation for interacting many-body quantum systems faces computational challenges due to exponential scaling with system size. This complexity limits the study of important phenomena in materials science and physics. We develop an Artificial Neural Network (ANN)-driven algorithm to simulate fermionic systems on lattices. Our method uses Pauli matrices to represent quantum states, incorporates Markov Chain Monte Carlo sampling, and leverages an adaptive momentum optimizer. We demonstrate the algorithm's accuracy by simulating the Heisenberg Hamiltonian on a one-dimensional lattice, achieving results with an error in the order of
10−4 compared to exact diagonalization. Furthermore, we successfully model a magnetic phase transition in a two-dimensional lattice under an applied magnetic field. Importantly, our approach avoids the sign problem common to traditional Fermionic Monte Carlo methods, enabling the investigation of frustrated systems. This work demonstrates the potential of ANN-based algorithms for efficient simulation of complex quantum systems, opening avenues for discoveries in condensed matter physics and materials science.
Solving the ground state of quantum many-body systems remains a fundamental challenge in physics and chemistry. Recent advancements in quantum hardware have opened new avenues for addressing this challenge. Inspired by the quantum-enhanced Markov chain Monte Carlo (QeMCMC) algorithm, which was originally designed for sampling the Boltzmann distribution of classical spin models using quantum computers, we introduce a quantum-assisted variational Monte Carlo (QA-VMC) algorithm for solving the ground state of quantum many-body systems by adapting QeMCMC to sample the distribution of a (neural-network) wave function in VMC. The central question is whether such a quantum-assisted proposal can potentially offer a computational advantage over classical methods. Through numerical investigations for the Fermi-Hubbard model and molecular systems, we demonstrate that the quantum-assisted proposal exhibits larger absolute spectral gaps and reduced autocorrelation times compared to conventional classical proposals, leading to more efficient sampling and faster convergence to the ground state in VMC as well as a more accurate and precise estimation of physical observables. This advantage is especially pronounced for specific parameter ranges, where the ground-state configurations are more concentrated in some configurations separated by large Hamming distances. Our results underscore the potential of quantum-assisted algorithms to enhance classical variational methods for solving the ground state of quantum many-body systems.
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.
We introduce a variational wave function based on neural-network quantum states (NQS) to study lattice systems whose local Hilbert space contains both spin and fermionic degrees of freedom. Our approach is based on the use of the transformer architecture, which can naturally handle composite local Hilbert spaces through a tokenization procedure closely inspired by techniques from natural language processing. The key methodological innovation is that the neural network predicts a set of fermionic orbitals that depend on the spin configuration in a backflow-inspired manner. We apply the method to the one-dimensional ancilla layer model, consisting of a chain of mobile spin- 1 / 2 fermions coupled to a two-leg spin- 1 / 2 ladder. For open boundary conditions, we achieve excellent quantitative agreement with density matrix renormalization group results across the full range of parameters considered. We find a phase in which the chain forms an effectively decoupled Luttinger liquid (LL), and an LL * phase with a distinct Fermi wave vector in which the mobile fermions are Kondo screened by one leg of the ladder, while the other leg forms the critical Bethe spin liquid. The LL * is the analog of the FL * phase in two dimensions proposed to describe the cuprate pseudogap. We also find a Luther-Emery phase, where the LL * state becomes unstable toward the formation of a spin gap. The transformer ansatz maintains comparable accuracy for periodic boundary conditions, where tensor-network methods are computationally more demanding. Moreover, we show that the approach extends naturally to fully periodic two-dimensional clusters, a regime beyond the reach of current tensor-network methods. Together, these findings establish transformer-based NQS as an accurate and scalable variational framework for correlated lattice systems with composite local Hilbert spaces and highlight its potential for studying higher-dimensional models where boundary effects and heterogeneous local structures pose significant challenges.
Deep neural networks have been very successful as highly accurate wave function Ansätze for variational Monte Carlo calculations of molecular ground states. We present an extension of one such Ansatz, FermiNet, to calculations of the ground states of periodic Hamiltonians, and study the homogeneous electron gas. FermiNet calculations of the ground-state energies of small electron gas systems are in excellent agreement with previous initiator full configuration interaction quantum Monte Carlo and diffusion Monte Carlo calculations. We investigate the spin-polarized homogeneous electron gas and demonstrate that the same neural network architecture is capable of accurately representing both the delocalized Fermi liquid state and the localized Wigner crystal state. The network converges on the translationally invariant ground state at high density and spontaneously breaks the symmetry to produce the crystalline ground state at low density, despite being given no a priori knowledge that a phase transition exists.
Strongly correlated molecules often contain dense manifolds of low-lying spin states, making total-spin symmetry essential for predictive electronic-structure theory. Neural-network quantum states provide flexible variational wavefunctions, but commonly used fermionic architectures do not enforce this symmetry and can therefore converge to spin-contaminated states with misleading energies and properties. Here we introduce a spin-adapted neural-network backflow (SA-NNBF) ansatz in second quantization, which combines configuration-dependent spatial orbitals with a compressed spin eigenfunction. A projected tensor compression scheme for spin eigenfunctions and a particle-hole representation make variational Monte Carlo calculations with SA-NNBF practical for active spaces containing more than one hundred electrons. Across hydrogen chains and iron-sulfur clusters, SA-NNBF eliminates spin contamination and consistently achieves lower variational energies than standard NNBF with a comparable number of parameters. For the CAS(113e,76o) active-space model of FeMoco, SA-NNBF yields a highly compact spin-adapted variational state, achieving an energy competitive with recent spin-adapted DMRG calculations at bond dimension
D=10000 while using orders of magnitude fewer parameters. Our work establishes a general framework for developing spin-symmetry-preserving neural-network quantum states for chemically realistic strongly correlated electrons.