Searching via walking: How to find a marked subgraph of a graph using quantum walks

TL;DR

This paper demonstrates how quantum walks, specifically scattering walks, can efficiently locate marked subgraphs within complete graphs, achieving quadratic speedup over classical methods through a quantum circuit implementation.

Contribution

It introduces a quantum walk-based method for searching marked subgraphs, with a practical implementation approach using quantum circuits and oracles.

Findings

Quantum walk localizes on the subgraph in O(N/K) steps.

Quantum search is quadratically faster than classical search.

Implementation details for quantum circuits and oracles provided.

Abstract

We show how a quantum walk can be used to find a marked edge or a marked complete subgraph of a complete graph. We employ a version of a quantum walk, the scattering walk, which lends itself to experimental implementation. The edges are marked by adding elements to them that impart a specific phase shift to the particle as it enters or leaves the edge. If the complete graph has N vertices and the subgraph has K vertices, the particle becomes localized on the subgraph in O(N/K) steps. This leads to a quantum search that is quadratically faster than a corresponding classical search. We show how to implement the quantum walk using a quantum circuit and a quantum oracle, which allows us to specify the resource needed for a quantitative comparison of the efficiency of classical and quantum searches -- the number of oracle calls.