On measurement-dependent variance in quantum neural networks

TL;DR

This paper investigates how measurement restrictions in quantum neural networks, such as in QCNNs, lead to increased prediction variance due to the eigenvalue spectrum of measured observables.

Contribution

It identifies the link between measurement support restrictions and increased variance, highlighting the role of eigenvalue spectrum in quantum machine learning.

Findings

Restricted measurements increase label prediction variance.

Eigenvalue spectrum of observables influences variance.

Measurement support impacts quantum neural network performance.

Abstract

Variational quantum circuits have become a widely used tool for performing quantum machine learning (QML) tasks on labeled quantum states. In some specific tasks or for specific variational ans\"atze, one may perform measurements on a restricted part of the overall input state. This is the case for, e.g., quantum convolutional neural networks (QCNNs), where after each layer of the circuit a subset of qubits of the processed state is measured or traced out, and at the end of the network one typically measures a local observable. In this work, we demonstrate that measuring observables with restricted support results in larger label prediction variance in regression QML tasks. We show that the reason for this is, essentially, the number of distinct eigenvalues of the observable one measures after the application of a variational circuit.