Search

Neural networks have shown to be a powerful tool to represent ground state of quantum many-body systems, including for fermionic systems. In this work, we introduce a framework for embedding lattice symmetries in Neural Slater-Backflow-Jastrow wavefunction ansatzes, and demonstrate how our model allows us to target the ground state and low-lying excited states. To capture the Hamiltonian symmetries, we introduce group-equivariant backflow transformations. We study the low-energy excitation spectrum of the t-V model on a square lattice away from half-filling, and find that our symmetry-aware backflow significantly improves the ground-state energies, and yields accurate low-lying excited states for up to 10x10 lattices. We additionally compute the two-point density-correlation function and the structure factor to detect the phase transition and determine the critical point. Finally, we quantify the variational accuracy of our model using the V-score.

Understanding the real-time evolution of many-electron quantum systems is essential for studying dynamical properties in condensed matter, quantum chemistry, and complex materials, yet it poses a significant theoretical and computational challenge. Our work introduces a variational approach for fermionic time-dependent wave functions, surpassing mean-field approximations by accurately capturing many-body correlations. Therefore, we employ time-dependent Jastrow factors and backflow transformations, which are enhanced through neural networks parameterizations. To compute the optimal time-dependent parameters, we utilize the time-dependent variational Monte Carlo technique and a new method based on Taylor-root expansions of the propagator, enhancing the accuracy of our simulations. The approach is demonstrated in three distinct systems. In all cases, we show clear signatures of many-body correlations in the dynamics. The results showcase the ability of our variational approach to accurately capture the time evolution, providing insight into the quantum dynamics of interacting electronic systems, beyond the capabilities of mean-field.

We introduce a message-passing-neural-network-based wave function Ansatz to simulate extended, strongly interacting fermions in continuous space. Symmetry constraints, such as continuous translation symmetries, can be readily embedded in the model. We demonstrate its accuracy by simulating the ground state of the homogeneous electron gas in three spatial dimensions at different densities and system sizes. With orders of magnitude fewer parameters than state-of-the-art neural-network wave functions, we demonstrate better or comparable ground-state energies. Reducing the parameter complexity allows scaling to $N=128$ electrons, previously inaccessible to neural-network wave functions in continuous space, enabling future work on finite-size extrapolations to the thermodynamic limit. We also show the Ansatz's capability of quantitatively representing different phases of matter., Comment: 13 pages, 5 figures

The continued development of computational approaches to many-body ground-state problems in physics and chemistry calls for a consistent way to assess its overall progress. In this work, we introduce a metric of variational accuracy, the V-score, obtained from the variational energy and its variance. We provide an extensive curated dataset of variational calculations of many-body quantum systems, identifying cases where state-of-the-art numerical approaches show limited accuracy, and future algorithms or computational platforms, such as quantum computing, could provide improved accuracy. The V-score can be used as a metric to assess the progress of quantum variational methods toward a quantum advantage for ground-state problems, especially in regimes where classical verifiability is impossible., Comment: 27 pages, 6 figures

Local Hamiltonians of fermionic systems on a lattice can be mapped onto local qubit Hamiltonians. Maintaining the locality of the operators comes at the expense of increasing the Hilbert space with auxiliary degrees of freedom. In order to retrieve the lower-dimensional physical Hilbert space that represents fermionic degrees of freedom, one must satisfy a set of constraints. In this work, we introduce quantum circuits that exactly satisfy these stringent constraints. We demonstrate how maintaining locality allows one to carry out a Trotterized time-evolution with constant circuit depth per time step. Our construction is particularly advantageous to simulate the time evolution operator of fermionic systems in d>1 dimensions. We also discuss how these families of circuits can be used as variational quantum states, focusing on two approaches: a first one based on general constant-fermion-number gates, and a second one based on the Hamiltonian variational ansatz where the eigenstates are represented by parametrized time-evolution operators. We apply our methods to the problem of finding the ground state and time-evolved states of the $t$-$V$ model.

Through the introduction of auxiliary fermions, or an enlarged spin space, one can map local fermion Hamiltonians onto local spin Hamiltonians, at the expense of introducing a set of additional constraints. We present a variational Monte-Carlo framework to study fermionic systems through higher-dimensional (>1D) Jordan-Wigner transformations. We provide exact solutions to the parity and Gauss-law constraints that are encountered in bosonization procedures. We study the $t$-$V$ model in 2D and demonstrate how both the ground state and the low-energy excitation spectra can be retrieved in combination with neural network quantum state ansatze.

Optimization of quantum states using the variational principle has recently seen an upsurge due to developments of increasingly expressive wave functions. In order to improve on the accuracy of the ans\"atze, it is a time-honored strategy to impose the systems' symmetries. We present an ansatz where global non-abelian symmetries are inherently embedded in its structure. We extend the model to incorporate lattice symmetries as well. We consider the prototypical example of the frustrated two-dimensional $J_1$-$J_2$ model on a square lattice, for which eigenstates have been hard to model variationally. Our novel approach guarantees that the obtained ground state will have total spin zero. Benchmarks on the 2D $J_1$-$J_2$ model demonstrate its state-of-the-art performance in representing the ground state. Furthermore, our methodology permits to find the wave functions of excited states with definite quantum numbers associated to the considered symmetries (including the non-abelian ones), without modifying the architecture of the network.

Although artificial neural networks have recently been proven to provide a promising new framework for constructing quantum many-body wave functions, the parameterization of a quantum wavefunction with nonabelian symmetries in terms of a Boltzmann machine inherently leads to biased results due to the basis dependence. We demonstrate that this problem can be overcome by sampling in the basis of irreducible representations instead of spins, for which the corresponding ansatz respects the nonabelian symmetries of the system. We apply our methodology to find the ground states of the one-dimensional antiferromagnetic Heisenberg (AFH) model with spin-half and spin-1 degrees of freedom, and obtain a substantially higher accuracy than when using the $s_z$-basis as input to the neural network. The proposed ansatz can target excited states, which is illustrated by calculating the energy gap of the AFH model. We also generalize the framework to the case of anyonic spin chains.