k-Contextuality as a Heuristic for Memory Separations in Learning

TL;DR

This paper introduces strong k-contextuality as a new measure to identify problems difficult for classical models but potentially easier for quantum models, supported by theoretical proofs and empirical algorithms.

Contribution

It defines strong k-contextuality, proves its relevance to classical memory limitations, and develops algorithms to estimate it, predicting classical-quantum performance differences.

Findings

Strong k-contextuality correlates with classical model limitations.

Algorithms effectively estimate contextuality in sequential data.

Empirical results show predictive power for classical vs. quantum model performance.

Abstract

Classical machine learning models struggle with learning and prediction tasks on data sets exhibiting long-range correlations. Previously, the existence of a long-range correlational structure known as contextuality was shown to inhibit efficient classical machine learning representations of certain quantum-inspired sequential distributions. Here, we define a new quantifier of contextuality we call strong k-contextuality, and prove that any translation task exhibiting strong k-contextuality is unable to be represented to finite relative entropy by a classical streaming model with fewer than k latent states. Importantly, this correlation measure does not induce a similar resource lower bound for quantum generative models. Using this theory as motivation, we develop efficient algorithms which estimate our new measure of contextuality in sequential data, and empirically show that this estimate is a good predictor for the difference in performance of resource-constrained classical and quantum Bayesian networks in modeling the data. Strong k-contextuality thus emerges as a measure to help identify problems that are difficult for classical computers, but may not be for quantum computers.