Multimarked Spatial Search by Continuous-Time Quantum Walk

TL;DR

This paper presents a general framework for analyzing the complexity of continuous-time quantum walk-based spatial search on arbitrary graphs, providing a method to determine optimal running times and success probabilities.

Contribution

It introduces a systematic approach to compute the spectral properties of the Hamiltonian for quantum walks on any graph, enabling precise complexity analysis.

Findings

Optimal search time is $O(\sqrt{N})$ on Johnson graphs with fixed diameter.

Success probability approaches 1 asymptotically.

Framework applicable to various graph structures for quantum spatial search.

Abstract

The quantum-walk-based spatial search problem aims to find a marked vertex using a quantum walk on a graph with marked vertices. We describe a framework for determining the computational complexity of spatial search by continuous-time quantum walk on arbitrary graphs by providing a recipe for finding the optimal running time and the success probability of the algorithm. The quantum walk is driven by a Hamiltonian derived from the adjacency matrix of the graph modified by the presence of the marked vertices. The success of our framework depends on the knowledge of the eigenvalues and eigenvectors of the adjacency matrix. The spectrum of the Hamiltonian is subsequently obtained from the roots of the determinant of a real symmetric matrix $M$, the dimensions of which depend on the number of marked vertices. The eigenvectors are determined from a basis of the kernel of $M$. We show each step of the framework by solving the spatial searching problem on the Johnson graphs with a fixed diameter and with two marked vertices. Our calculations show that the optimal running time is $O(\sqrt{N})$ with an asymptotic probability of $1+o(1)$, where $N$ is the number of vertices.