Spatial Search on Graphs with Multiple Targets using Flip-flop Quantum Walk

TL;DR

This paper analyzes the spectral properties of flip-flop quantum walks on regular graphs and demonstrates their quadratic speedup over classical walks in spatial search tasks with multiple targets, providing optimal or near-optimal complexity results.

Contribution

It introduces a simplified quantum spatial search algorithm with proven optimality in certain graph classes and analyzes its spectral and complexity properties.

Findings

Quantum walk is quadratically faster than classical walk.

The algorithm's oracle complexity is optimal for certain graph classes.

Provides bounds on classical hitting times for regular graphs.

Abstract

We analyse the eigenvalue and eigenvector structure of the flip-flop quantum walk on regular graphs, explicitly demonstrating how it is quadratically faster than the classical random walk. Then we use it in a controlled spatial search algorithm with multiple target states, and determine the oracle complexity as a function of the spectral gap and the number of target states. The oracle complexity is optimal as a function of the graph size and the number of target states, when the spectral gap of the adjacency matrix is $\Theta(1)$. It is also optimal for spatial search on $D>4$ dimensional hypercubic lattices. Otherwise it matches the best result available in the literature, with a much simpler algorithm. Our results also yield bounds on the classical hitting time of random walks on regular graphs, which may be of independent interest.