Search algorithm on strongly regular graphs based on scattering quantum walk☆☆Project supported by the National Natural Science Foundation of China (Grant Nos. 61502101 and 61170321), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20140651), and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110092110024).

Abstract

Janmark, Meyer, and Wong showed that continuous-time quantum walk search on known families of strongly regular graphs (SRGs) with parameters achieves full quantum speedup. The problem is reconsidered in terms of scattering quantum walk, a type of discrete-time quantum walks. Here, the search space is confined to a low-dimensional subspace corresponding to the collapsed graph of SRGs. To quantify the algorithm's performance, we leverage the fundamental pairing theorem, a general theory developed by Cottrell for quantum search of structural anomalies in star graphs. The search algorithm on the SRGs with k scales as N satisfies the theorem, and results can be immediately obtained, while search on the SRGs with k scales as does not satisfy the theorem, and matrix perturbation theory is used to provide an analysis. Both these cases can be solved in time steps with a success probability close to 1. The analytical conclusions are verified by simulation results on two SRGs. These examples show that the formalism on star graphs can be applied more generally.