Learning the ground state of a non-stoquastic quantum Hamiltonian in a rugged neural network landscape

TL;DR

This paper explores the use of neural network-based variational wave-functions to approximate the ground state of a complex non-stoquastic quantum Hamiltonian, highlighting the challenges posed by a rugged energy landscape.

Contribution

It introduces a neural network ansatz with separate amplitude and phase networks and analyzes the impact of the energy landscape's ruggedness on learning the ground state.

Findings

Rugged energy landscape is the main obstacle in ground state approximation.

Neural network expressivity and sampling are not the primary limitations.

Mitigation strategies improve variational energy to recent neural network benchmarks.

Abstract

Strongly interacting quantum systems described by non-stoquastic Hamiltonians exhibit rich low-temperature physics. Yet, their study poses a formidable challenge, even for state-of-the-art numerical techniques. Here, we investigate systematically the performance of a class of universal variational wave-functions based on artificial neural networks, by considering the frustrated spin-$1/2$ $J_1-J_2$ Heisenberg model on the square lattice. Focusing on neural network architectures without physics-informed input, we argue in favor of using an ansatz consisting of two decoupled real-valued networks, one for the amplitude and the other for the phase of the variational wavefunction. By introducing concrete mitigation strategies against inherent numerical instabilities in the stochastic reconfiguration algorithm we obtain a variational energy comparable to that reported recently with neural networks that incorporate knowledge about the physical system. Through a detailed analysis of the individual components of the algorithm, we conclude that the rugged nature of the energy landscape constitutes the major obstacle in finding a satisfactory approximation to the ground state wavefunction, and prevents learning the correct sign structure. In particular, we show that in the present setup the neural network expressivity and Monte Carlo sampling are not primary limiting factors.