Scaling Laws for Neural-Network Quantum States

2026-06-01

AI Summary

This work investigates whether scaling laws analogous to those in modern neural networks govern the learning complexity of strongly correlated quantum many-body systems. By combining Transformer-based variational quantum states with large-scale variational Monte Carlo methods, the study examines the relationship between approximation accuracy and computational effort in representing the ground states of the J₁–J₂ Heisenberg model on triangular and square lattices. The authors establish, for the first time in neural-network quantum states, a system-size-independent power-law scaling: the V-score decays as a power law with increasing computational cost, and data across different system sizes collapse onto a single universal curve after rescaling. Notably, the scaling exponent systematically decreases with enhanced frustration, quantitatively linking magnetic frustration to the representational difficulty of the ground state and providing a general benchmark framework for variational wave functions.

Abstract

Scaling laws, the power-law relations between loss, architecture size, and compute observed in modern neural networks, offer a quantitative way to characterize the complexity of a learning problem, with the exponent governing the decay of the loss reflecting how rapidly additional resources translate into improved accuracy, and thus how hard the target is to learn. Whether an analogous framework can characterize the complexity of physical problems remains open. We address this question for Neural-Network Quantum States, a leading variational approach for strongly correlated quantum many-body systems. Using transformer wave functions to approximate ground states of the $J_1$-$J_2$ Heisenberg model on triangular and square lattices with up to $20\times 20$ sites, we find that the $V$-score, a measure of accuracy of a variational state, decays as a power law in training compute. Under an appropriate rescaling of compute, results for different system sizes collapse onto a single curve, analogous to scaling collapse in critical phenomena. The resulting power law is, to a good approximation, independent of the number of sites, showing that the transformer Ansatz is size-consistent for the systems considered. The exponent decreases systematically with frustration, identifying it as a quantitative measure of representational difficulty of the ground state and establishing scaling laws as a general framework for benchmarking variational ansätze.

Problem

• Scaling Laws
• Neural-Network Quantum States
• Quantum Many-Body Systems
• Representational Difficulty
• Variational Ansatz

Innovation

• scaling laws
• neural-network quantum states
• transformer wave functions
• size consistency
• frustrated quantum systems