Expressibility of neural quantum states: a Walsh-complexity perspective

TL;DR

This paper introduces Walsh complexity as a measure of wavefunction spread over parity patterns, analyzing neural quantum states' expressibility and the depth needed for neural networks to efficiently represent complex states.

Contribution

It defines Walsh complexity and demonstrates its role in understanding the limitations and capabilities of shallow neural networks in quantum state representation.

Findings

States with uniform Walsh spectrum require exponential Walsh complexity.

Shallow networks cannot generate high Walsh complexity in the tame regime.

Depth scales logarithmically with system size for successful fitting with polynomial activations.

Abstract

Neural quantum states are powerful variational wavefunctions, but it remains unclear which many-body states can be represented efficiently by modern additive architectures. We introduce Walsh complexity, a basis-dependent measure of how broadly a wavefunction is spread over parity patterns. States with an almost uniform Walsh spectrum require exponentially large Walsh complexity from any good approximant. We show that shallow additive feed-forward networks cannot generate such complexity in the tame regime, e.g. polynomial activations with subexponential parameter scaling. As a concrete example, we construct a simple dimerized state prepared by a single layer of disjoint controlled-$Z$ gates. Although it has only short-range entanglement and a simple tensor-network description, its Walsh complexity is maximal. Full-cube fits across system size and depth are consistent with the complexity bound: for polynomial activations, successful fitting appears only once depth reaches a logarithmic scale in $N$, whereas activation saturation in $\tanh$ produces a sharp threshold-like jump already at depth $3$. Walsh complexity therefore provides an expressibility axis complementary to entanglement and clarifies when depth becomes an essential resource for additive neural quantum states.