Information-Theoretic Scaling Laws of Neural Quantum States

TL;DR

This paper derives an information-theoretic scaling law for autoregressive neural quantum states, linking the wavefunction's mutual information to the neural network's complexity, and validates it through numerical experiments across various quantum tasks.

Contribution

It introduces a rigorous scaling law connecting mutual information with neural network capacity for quantum states, including explicit formulas for stabilizer states and empirical validation.

Findings

Scaling law accurately predicts neural network size requirements

Recurrent and Transformer architectures exhibit different scaling behaviors

Autoregressive basis ordering significantly impacts model efficiency

Abstract

We establish an information-theoretic scaling law for generic autoregressive neural quantum states, determined by the middle-cut mutual information of the wavefunction amplitude. By formalizing the virtual bond as an effective information channel across a sequence bipartition, we rigorously prove that exact autoregressive representation of a quantum state requires the virtual-bond dimension to scale with the amplitude mutual information. For stabilizer-state families, we show that this law yields an explicit, analytical rank formula. Applying this framework across quantum-state tomography, ground-state and finite-temperature learning, our numerical experiments expose precise exponent matching, architecture-dependent scaling differences between recurrent and Transformer neural quantum state, and the critical role of autoregressive basis ordering. These results establish a rigorous physical link between the intrinsic structure of a quantum many-body state and the corresponding neural-network capacity required for its faithful representation.