Unstructured Search by Random and Quantum Walk

TL;DR

This paper explores how classical and quantum walks can be used to solve unstructured search problems, providing new insights and detailed derivations of their efficiencies and behaviors.

Contribution

It offers a comprehensive, pedagogical analysis of classical and quantum walks for spatial search, including some new results on their convergence and performance.

Findings

Random walks converge to similar evolution, taking N ln(1/ε) time.

Discrete-time quantum walk takes approximately π√N/2√2 steps.

Continuous-time quantum walk takes approximately π√N/2 steps.

Abstract

The task of finding an entry in an unsorted list of $N$ elements famously takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$ queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large $N$, the random walks converge to the same evolution, both taking $N \ln(1/\epsilon)$ time to reach a success probability of $1-\epsilon$. In contrast, the discrete-time quantum walk asymptotically takes $\pi\sqrt{N}/2\sqrt{2}$ timesteps to reach a success probability of $1/2$, while the continuous-time quantum walk takes $\pi\sqrt{N}/2$ time to reach a success probability of $1$.