Revisiting quantum walk advantages: A mean hitting time perspective

TL;DR

The paper addresses the limitation of using mean squared displacement (MSD) to compare quantum and classical walks by focusing on mean hitting time (MHT), which has clear operational meaning for search tasks and is sensitive to non-Gaussian distributions inherent in quantum walks. It analyzes a discrete-time quantum walk (DTQW) on a chain with two detectors and employs the Krovi-Brun formalism to compute MHT, extending the framework to stochastic resetting via CPTP Kraus maps to reveal potential quantum advantages. A key result is that, under symmetric initial conditions with two detectors, quantum and classical MHT are identical, challenging the view that quantum walks universally outperform classical ones in these settings; however, resetting can selectively reduce the quantum MHT, signaling a quantum advantage through quasi-momentum redistribution, which degrades with noise and can vanish at high decoherence. The work highlights that different metrics capture different aspects of quantum-classical speedups and proposes a reset-based MHT criterion as a practical signature of quantum behavior for benchmarking quantum walk implementations on noisy devices.

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.

Figures (7)

• Figure 1: The modified Galton board experiment scheme. The particles (balls) are falling from the top and bouncing of the wooden pins either to the left or to the right. At the ends of the board we have shelfs which trap the particles acting as detectors. They measure at what time the particle went to the target position. Repeating this experiment with many particle will give us the average time of the particle being measured at target positions so the MHT.
• Figure 2: The spatial probability distributions of the classical (blue) and quantum (red) walks on chain.
• Figure 3: The MSD over time for classical (blue) and quantum (red) walk starting from initial gaussian distributions with widths $\sigma = 10$ (dots and diamonds) and $\sigma =0$ (stars and crosses). Additionally the quantum walk has been initially with "symmetric" coin state i.e. $\frac{1}{\sqrt{2}}(|{+1}\rangle+i|{-1}\rangle)$.
• Figure 4: The probability distribution of particle during quantum walk evolution starting from initial state $|{\Psi}\rangle = N\sum_x e^{-(\frac{x}{2\sigma})^2}|{x,+1}\rangle$.
• Figure 5: Quantum counterpart of the modified Galton board experiment scheme. Photons are shot from the top onto beamsplitters. At the ends of the board we have detectors. They measure at what time the photon went to the target position. Repeating this experiment with many particle will give us the average time of the photon being measured at target positions so the MHT.