Amplitude Ratios and Neural Network Quantum States

TL;DR

This paper explores the access models for neural network quantum states, introduces amplitude ratio access, compares it with existing models, and constructs a small neural network that fails to encode a valid wavefunction.

Contribution

It defines and analyzes the amplitude ratio access model, comparing its strength to existing models, and constructs a minimal neural network state that is invalid.

Findings

Amplitude ratio access is stronger than sample access.

Amplitude ratio access is weaker than sample and query access but retains many capabilities.

Constructed a 3-node neural network that does not encode a valid wavefunction.

Abstract

Neural Network Quantum States (NQS) represent quantum wavefunctions by artificial neural networks. Here we study the wavefunction access provided by NQS defined in [Science, \textbf{355}, 6325, pp. 602-606 (2017)] and relate it to results from distribution testing. This leads to improved distribution testing algorithms for such NQS. It also motivates an independent definition of a wavefunction access model: the amplitude ratio access. We compare it to sample and sample and query access models, previously considered in the study of dequantization of quantum algorithms. First, we show that the amplitude ratio access is strictly stronger than sample access. Second, we argue that the amplitude ratio access is strictly weaker than sample and query access, but also show that it retains many of its simulation capabilities. Interestingly, we only show such separation under computational assumptions. Lastly, we use the connection to distribution testing algorithms to produce an NQS with just three nodes that does not encode a valid wavefunction and cannot be sampled from.