Exceptional Quantum Walk Search on the Cycle

TL;DR

This paper identifies exceptional cases where quantum walks on cycles do not provide speedups in search, but demonstrates a different scenario where quantum sampling still offers significant advantages over classical methods.

Contribution

It proves that cycles with any marked vertex arrangement are exceptional cases for quantum walk speedups and constructs a problem showing quantum sampling can outperform classical hitting times.

Findings

Cycles with any marked vertices are exceptional for quantum walk speedups.

Quantum sampling can achieve arbitrary speedup over classical hitting time.

Speedup is roughly quadratic when considering mixing time instead of hitting time.

Abstract

Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk's hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk's random sampling yields an arbitrary speedup in query complexity over the classical random walk's hitting time. In this context, however, the mixing time to prepare the initial uniform state is a more suitable comparison than the hitting time, and then the speedup is roughly quadratic.