Efficiency of neural-network state representations of one-dimensional quantum spin systems

TL;DR

This paper analyzes the expressibility and complexity of neural-network quantum state representations, especially RBMs, for 1D quantum spin systems, proposing a new paradigm for complexity analysis and suggesting broad applicability.

Contribution

It introduces a class of LRFD RBM states with bounded errors, providing a framework for complexity analysis and conjecturing exact representation of ground states for various quantum systems.

Findings

LRFD RBMs can approximate many 1D quantum systems with polynomial complexity

Ground states of diverse quantum systems may be exactly represented by LRFD RBMs

A new paradigm for complexity analysis of long-range RBMs is proposed

Abstract

Neural-network state representations of quantum many-body systems are attracting great attention and more rigorous quantitative analysis about their expressibility and complexity is warranted. Our analysis of the restricted Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems provides new insight into their computational complexity. We define a class of long-range-fast-decay (LRFD) RBM states with quantifiable upper bounds on truncation errors and provide numerical evidence for a large class of 1D quantum systems that may be approximated by LRFD RBMs of at most polynomial complexities. These results lead us to conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them, even in cases where other state representations become less efficient. At last, we provide the relations between multiple typical state manifolds. Our work proposes a paradigm for doing complexity analysis for generic long-range RBMs which naturally yields a further classification of this manifold. This paradigm and our characterization of their nonlocal structures may pave the way for understanding the natural measure of complexity for quantum many-body states described by RBMs and are generalizable for higher-dimensional systems and deep neural-network quantum states.