Variational Monte Carlo with Neural Network Quantum States for Yang-Mills Matrix Model

TL;DR

This paper demonstrates that neural network quantum states combined with variational Monte Carlo can accurately approximate the ground state energy of the bosonic SU(N) Yang-Mills matrix model at strong coupling, validated against lattice Monte Carlo results.

Contribution

It introduces a neural autoregressive flow architecture as an ansatz for variational Monte Carlo to study the Yang-Mills matrix model at strong coupling, achieving accurate results for larger system sizes.

Findings

Neural network states reproduce ground state energies for N=2 and 3.

The approach's accuracy improves with larger network width.

More computational resources are needed for N=4 to achieve similar accuracy.

Abstract

We apply the variational Monte Carlo method based on neural network quantum states, using a neural autoregressive flow architecture as our ansatz, to determine the ground state wave function of the bosonic SU($N$) Yang-Mills-type two-matrix model at strong coupling. Previous literature hinted at the inaccuracy of such an approach at strong coupling. In this work, the accuracy of the results is tested using lattice Monte Carlo simulations: we benchmark the expectation value of the energy of the ground state for system sizes $N$ that are beyond brute-force exact diagonalization methods. We observe that the variational method with neural network states reproduces the right ground state energy when the width of the network employed in this work is sufficiently large. We confirm that the correct result is obtained for $N=2$ and $3$, while obtaining a precise value for $N=4$ requires more resources than the amount available for this work.