Search algorithm on strongly regular graphs based on scattering quantum walk

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Journal Article
Chinese Physics B, 2017, 26 (1)
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© 2017 Chinese Physical Society and IOP Publishing Ltd. Janmark, Meyer, and Wong showed that continuous-time quantum walk search on known families of strongly regular graphs (SRGs) with parameters (N,k,λ,m) 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 algorithms 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 √N does not satisfy the theorem, and matrix perturbation theory is used to provide an analysis. Both these cases can be solved in O(√N) 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.
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