Spatial Search on Johnson Graphs by Discrete-Time Quantum Walk

TL;DR

This paper investigates the efficiency of quantum walks for spatial search on Johnson graphs, demonstrating that a quantum approach can achieve a success probability of 1/2 after a specific number of steps, improving understanding of quantum search algorithms.

Contribution

The paper introduces a quantum walk-based method for spatial search on Johnson graphs and calculates the asymptotic success probability and complexity, leveraging the graph's symmetry.

Findings

Success probability of 1/2 after π√N/(2√2) steps

Quantum walk approach improves search efficiency on Johnson graphs

Invariant subspace simplifies complexity analysis

Abstract

The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) The walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker's position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex- and distance-transitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is $1/2$ after taking $\pi\sqrt N/(2\sqrt 2)$ steps, where $N$ is the number of vertices of the Johnson graph.