Neural network wave functions and the sign problem

Attila Szabó,Claudio Castelnovo

doi:10.1103/physrevresearch.2.033075

Abstract

This paper proposes a neural network architecture and learning protocol to solve the problem of convergence to ground states with a nontrivial sign structure. The authors apply their scheme to conventional antiferromagnets and show that, while it works well on those systems, it is unable to find ground states on frustrated magnets, finding instead low-energy states that exhibit the unfrustrated Marshall sign rule

Highlights

Over the last decade, machine learning has had a profound impact on most aspects of life as well as the physical sciences [1]
neural quantum states (NQS) approaches are fundamentally appealing because both restricted Boltzmann machine (RBM) [8] and deep neural networks [9] can represent almost any function accurately, without a priori limitations like the Monte Carlo sign problem [10] or the area-law entanglement of tensor network (TN) [11,12,13]
Since S2, P, and the point-group symmetry operators all commute with the Hamiltonian (3), the true ground state is an eigenstate of the former two, and transforms according to precisely one irrep of the latter; ground states of Heisenberg antiferromagnet (HAFM) are normally singlets (S2 = 0) and have even parity: Deviations from these expectations can be used as a quantitative test of the converged wave functions

Summary

INTRODUCTION

Machine learning has had a profound impact on most aspects of life as well as the physical sciences [1]. While NQS Ansätze can in principle represent nontrivial sign structures, learning them appears to pose significant challenges, especially in frustrated systems This appears less pointedly in RBMs and other shallow, fully connected architectures, which are able to reach low variational energies even for Hamiltonians with a severe sign problem [3,15,16,17,18,19,20]. MSR, even though the true ground state is expected to deviate from it significantly The existence of such “MSR-like” lowenergy variational states, and the ease and stability with which the VMC algorithm homes in on them, highlight the risks of using the energy as the only criterion for assessing the accuracy of variational wave functions and may explain the poor generalisation properties of supervised NQS learning in frustrated regimes [16,19]. We suggest potential improvements toward the end of this work.

OUR APPROACH

NUMERICAL EXPERIMENTS

DISCUSSION

Neural network architectures

Optimization protocol

Observable estimation