# Restricted Boltzmann Machines for Quantum States with Nonabelian or Anyonic Symmetries

## TL;DR

This paper introduces a symmetry-respecting neural network approach using basis sampling in irreducible representations to accurately model quantum states with nonabelian and anyonic symmetries, improving over previous basis-dependent methods.

## Contribution

It proposes a basis sampling method in irreducible representations for neural network quantum states, overcoming basis dependence issues in nonabelian symmetry modeling.

## Key findings

- Achieved higher accuracy in ground state calculations of the AFH model.
- Successfully targeted excited states and calculated energy gaps.
- Generalized the framework to anyonic spin chains.

## Abstract

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.