# Expressibility of neural quantum states: a Walsh-complexity perspective

arXiv: https://arxiv.org/abs/2604.03294 · HTML: https://mlsanity.com/paper/2604.03294/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 24 references that are in the map.
- First posted on arXiv 27 Mar 2026.

## Builds on (most cited first, 2 of 2)

- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Neural tensor contractions and the expressive power of deep neural quantum states](https://mlsanity.com/paper/2103.10293/) (2021)

## Abstract (arXiv)

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.