Kernel Matrix Completion for Offline Quantum-Enhanced Machine Learning

TL;DR

This paper introduces a classical matrix completion method for quantum kernel matrices in quantum machine learning, enabling efficient incorporation of new data without costly quantum computations, with empirical validation on real-world data.

Contribution

It proposes a novel classical matrix completion approach for quantum kernels, addressing data update challenges in quantum ML workflows and analyzing its theoretical and empirical performance.

Findings

Quantum kernel matrices can be completed with minimal sample complexity.

Completion error degrades gracefully with finite-sampling noise.

Kernel matrix rank depends weakly on the quantum feature map expressibility.

Abstract

Enhancing classical machine learning (ML) algorithms through quantum kernels is a rapidly growing research topic in quantum machine learning (QML). A key challenge in using kernels -- both classical and quantum -- is that ML workflows involve acquiring new observations, for which new kernel values need to be calculated. Transferring data back-and-forth between where the new observations are generated & a quantum computer incurs a time delay; this delay may exceed the timescales relevant for using the QML algorithm in the first place. In this work, we show quantum kernel matrices can be extended to incorporate new data using a classical (chordal-graph-based) matrix completion algorithm. The minimal sample complexity needed for perfect completion is dependent on matrix rank. We empirically show that (a) quantum kernel matrices can be completed using this algorithm when the minimal sample complexity is met, (b) the error of the completion degrades gracefully in the presence of finite-sampling noise, and (c) the rank of quantum kernel matrices depends weakly on the expressibility of the quantum feature map generating the kernel. Further, on a real-world, industrially-relevant data set, the completion error behaves gracefully even when the minimal sample complexity is not reached.