Quantum Markov Chains: Hub-Pruned Estimation for Fashion Recommenders

TL;DR

This paper explores how shallow quantum circuits can model fashion recommendation Markov chains, showing that removing hub nodes improves quantum-classical agreement and offers practical benchmarking for near-term quantum experiments.

Contribution

It introduces a hub-pruned quantum estimation method for recommendation dynamics, demonstrating improved accuracy and providing a reproducible benchmarking protocol.

Findings

Hub pruning reduces divergence metrics by about half.

State fidelities remain close to unity despite pruning.

Hub pruning enhances quantum-classical agreement at shallow depths.

Abstract

We investigate whether shallow quantum circuits can accurately reproduce the short-horizon dynamics of discrete-time Markov chains derived from fashion electronic-commerce recommendation links. Transition operators are compiled into block-encoded circuits and iterated using fixed-point oblivious amplitude amplification, and amplitude-encoded marginals are used to estimate the classical push-forward. Empirically, colour categories such as black and, to a lesser extent, white function as high-degree hubs that dominate probability flow. Consequently, we assess three chain variants: the full network including all colours, a network without black, and a network without black and white, to quantify the effect of hub pruning under realistic circuit depths and measurement budgets. Across networks aggregated from multiple retailers, hub pruning consistently improves quantum and classical agreement at shallow depth; total-variation distance and Kullback-Leibler divergence typically decrease by approximately a factor of two relative to the full network, while state fidelities remain close to unity. A bias-and-contraction analysis explains these gains through reduced cross-terms and an effectively widened spectral gap. The results identify hub-pruned block-encodings as a practical heuristic for near-term experiments on recommendation dynamics using small quantum registers, and they provide a reproducible benchmarking protocol that reports total-variation distance, Kullback-Leibler divergence, and state fidelity as functions of circuit depth and prediction horizon.