Revisiting quantum walk advantages: A mean hitting time perspective

TL;DR

This paper explores mean hitting time as a new metric for comparing quantum and classical walks, revealing context-dependent advantages and effects of stochastic resetting, especially relevant for noisy quantum devices.

Contribution

It introduces mean hitting time as a complementary metric to MSD, analyzes quantum advantage under reset, and highlights the metric's relevance for noisy quantum walk implementations.

Findings

Quantum and classical walks have identical MHT for symmetric initial conditions.

Stochastic resetting can reduce MHT in quantum walks but not in classical walks.

Quantum advantage diminishes with noise, converging to classical behavior.

Abstract

The mean squared displacement has been widely used as the primary metric for comparing quantum and classical random walks, with quantum walks showing quadratic scaling versus linear scaling for classical walks. However, this comparison may not capture the full picture: while the mean squared displacement is well-suited for Gaussian distributions, quantum walk distributions exhibit distinctly non-Gaussian features. We propose that the mean hitting time offers a complementary perspective with clear operational meaning for search algorithms. Through analytical calculations, we show that quantum and classical walks yield identical MHT for symmetric initial conditions with two detectors, suggesting that the apparent quantum advantage seen in MSD comparisons may be context-dependent. Interestingly, introducing stochastic resetting reveals new dynamics. We demonstrate analytically that quantum walks can achieve reduced MHT under stochastic reset through quasi-momentum redistribution, while classical walks see no benefit. This quantum advantage naturally degrades with noise, the quantum walk converges to classical behavior. We suggest that MHT reduction under stochastic reset can serve as an additional signature of quantum behavior, particularly useful for characterizing quantum walk implementations on noisy quantum devices. Our results indicate that different metrics can reveal different aspects of quantum-classical comparisons in walk-based algorithms.