Neural Network Ground State from the Neural Tangent Kernel Perspective: The Sign Bias

TL;DR

This paper analyzes the limitations and effectiveness of neural networks in representing quantum ground states, focusing on the neural tangent kernel and basis dependence, especially regarding the sign problem.

Contribution

It provides a theoretical analysis of neural network convergence in quantum state representation, highlighting the impact of basis choice and sign bias on performance.

Findings

Neural tangent kernel analysis reveals basis-dependent convergence properties.

Stoquastic Hamiltonians are more effectively represented by neural networks.

Sign bias influences the success of neural network approaches in quantum problems.

Abstract

Neural networks has recently attracted much interest as useful representations of quantum many body ground states, which might help address the infamous sign problem. Most attention was directed at their representability properties, while possible limitations on finding the desired optimal state have not been suitably explored. By leveraging well-established results applicable in the context of infinite width, specifically regarding the renowned neural tangent kernel and conjugate kernel, a comprehensive analysis of the convergence and initialization characteristics of the method is conducted. We reveal the dependence of these characteristics on the interplay among these kernels, the Hamiltonian, and the basis used for its representation. We introduce and motivate novel performance metrics and explore the condition for their optimization. By leveraging these findings, we elucidate a substantial dependence of the effectiveness of this approach on the selected basis, demonstrating that so-called stoquastic Hamiltonians are more amenable to solution through neural networks than those suffering from a sign problem.