Quantum Search with Multiple Walk Steps per Oracle Query

TL;DR

This paper demonstrates that continuous-time quantum walks can outperform discrete-time walks in quantum search by performing multiple walk steps per oracle query, leading to a cubic speedup over classical random walks.

Contribution

It shows that allowing discrete-time quantum walks to perform multiple steps per query matches continuous-time performance and achieves a cubic speedup over classical walks, surpassing previous quadratic limits.

Findings

Continuous-time quantum walks outperform discrete-time walks in search.

Multiple walk steps per query enable cubic speedup over classical random walks.

First example of greater-than-quadratic quantum speedup over classical methods.

Abstract

We identify a key difference between quantum search by discrete- and continuous-time quantum walks: a discrete-time walk typically performs one walk step per oracle query, whereas a continuous-time walk can effectively perform multiple walk steps per query while only counting query time. As a result, we show that continuous-time quantum walks can outperform their discrete-time counterparts, even though both achieve quadratic speedups over their corresponding classical random walks. To provide greater equity, we allow the discrete-time quantum walk to also take multiple walk steps per oracle query while only counting queries. Then it matches the continuous-time algorithm's runtime, but such that it is a cubic speedup over its corresponding classical random walk. This yields the first example of a greater-than-quadratic speedup for quantum search over its corresponding classical random walk.