Expressibility of neural quantum states: a Walsh-complexity perspective
Provides a new theoretical framework for understanding when depth is necessary for neural quantum states, addressing a fundamental question in quantum machine learning.
The paper introduces Walsh complexity to measure the expressibility of neural quantum states, showing that shallow additive networks cannot represent states with high Walsh complexity. They construct a dimerized state with maximal Walsh complexity despite simple entanglement, and demonstrate that depth is essential for fitting such states.
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.