Bound on entanglement in neural quantum states

TL;DR

The paper addresses how much entanglement neural quantum states (NQS) can generate and seeks fundamental constraints analogous to the area law for matrix product states. They prove that feed-forward NQS with $k$ nonlinearities satisfy $S_A = O(k \log n)$ for any subregion $A$, with $\\mu \\le k+1$ affine features and under the assumption that the output function $\\mathcal{G}$ is analytic and bounded, enabling a polynomial approximation that yields the bound as $n \\to \\infty$. This result rules out volume-law entanglement for $k = O(1)$ and is demonstrated to be tight across several architectures (SN-NQS, MLP-NQS, T-NQS), including a Dicke-state construction showing $S_A \\sim (c/2) \\log n$ for $k = 1$. The work provides a universal, architecture-agnostic constraint on NQS expressivity, clarifying when sub-volume-law entanglement is inevitable and guiding the design of efficient variational representations, with future directions toward recurrent networks and relaxed analyticity assumptions.

Abstract

Variational wavefunctions offer a practical route around the exponential complexity of many-body Hilbert spaces, but their expressive power is often sharply constrained. Matrix product states, for instance, are efficient but limited to area law entangled states. Neural quantum states (NQS) are widely believed to overcome such limitations, yet little is known about their fundamental constraints. Here we prove that feed-forward neural quantum states acting on $n$ spins with $k$ scalar nonlinearities, under certain analyticity assumptions, obey a bound on entanglement entropy for any subregion: $S \leq c k\log n$, for a constant $c$. This establishes an NQS analog of the area law constraint for matrix product states and rules out volume law entanglement for NQS with $O(1)$ nonlinearities. We demonstrate analytically and numerically that the scaling with $n$ is tight for a wide variety of NQS. Our work establishes a fundamental constraint on NQS that applies broadly across different network designs, while reinforcing their substantial expressive power.

Figures (2)

• Figure 1: (a) Subregion von Neumann entanglement entropy $S_A$ for $n=22$ and (b) bipartite von Neumann entanglement entropy $S_{n/2}$ for the Dicke state, Eq. \ref{['eq:Dicke']}, on a logarithmic $x$-axis, corroborating analytically predicted scaling, respectively, up to finite-size effects. (c,d) Subregion and bipartite entanglement entropy for single-nonlinearity, multilayer perceptron, and transformer neural quantum states (SN-NQS, MLP-NQS, T-NQS) on a logarithmic $x$-axis with random parameters and shading indicating one standard deviation. The MLP-NQS has width $3$ and depth $2$. For the T-NQS, we chose patches of size $6$, stride $5$, 4 attention heads, and 2 layers. Each point was averaged over 20 trial wavefunctions.
• Figure 2: (a) Subregion entanglement entropy for CosNet states with $n=22$ qubits, averaged over 20 random initializations. The dashed black line is the Page value, i.e. the Haar-random state prediction. Consistent with (sub)logarithmic entanglement scaling for $k\ll n$ and appears volume law for $k\gg n$. (b) $S_{|A|=7}$ for varying $k$. Plots have logarithmic $x$-axes, shading indicates one standard deviation, and $(\sigma_a,\sigma_w)=(10,1)$.