Quantum Walk Search on Complete Multipartite Graph with Multiple Marked Vertices

TL;DR

This paper explores quantum walk search algorithms on complete multipartite graphs with multiple marked vertices, demonstrating quadratic speedup and robustness even with unknown marked vertices, supported by simulations and circuit implementations.

Contribution

It introduces a quantum walk search method on complete multipartite graphs with multiple marked vertices, achieving quadratic speedup and robustness, which was not previously studied.

Findings

Quadratic speedup achieved in search efficiency.

Robustness of quantum walk with unknown number of marked vertices.

Successful numerical simulation and circuit implementation.

Abstract

Quantum walk is a potent technique for building quantum algorithms. This paper examines the quantum walk search algorithm on complete multipartite graphs with multiple marked vertices, which has not been explored before. Two specific cases of complete multipartite graphs are probed in this paper, and in both cases, each set consists of an equal number of vertices. We employ the coined quantum walk model and achieve quadratic speedup with a constant probability of finding a marked vertex. Furthermore, we investigate the robust quantum walk of two cases and demonstrate that even with an unknown number of marked vertices, it is still possible to achieve a quadratic speedup compared to classical algorithms and the success probability oscillates within a small range close to 1. This work addresses the overcooking problem in quantum walk search algorithms on some complete multipartite graphs. We also provide the numerical simulation and circuit implementation of our quantum algorithm.