On the hitting times of quantum versus random walks

TL;DR

This paper introduces new classical and quantum hitting times, explores their relationships, and develops quantum algorithms for detection and finding problems with complexities tied to these hitting times.

Contribution

It defines new Monte Carlo type hitting times, establishes their relationships with existing definitions, and presents quantum algorithms leveraging these times for improved detection and finding tasks.

Findings

Quantum hitting time is proportional to the square root of classical hitting time.

For reversible ergodic Markov chains, quantum hitting time matches the order of classical hitting time's square root.

The new algorithms are simpler and potentially more efficient, with complexities related to the new quantum hitting times.

Abstract

In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Further, we prove that for any reversible ergodic Markov chain $P$, the quantum hitting time of the quantum analogue of $P$ has the same order as the square root of the classical hitting time of $P$. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. Finally, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem for the 2D grid. Extending his result, we show that for any state-transitive Markov chain with unique marked state, the quantum hitting time is of the same order for both the detection and finding problems.