Abstract
The Johnson graph J (n, k) is defined by n symbols, where vertices are k-element subsets of the symbols, and vertices are adjacent if they differ in exactly one symbol. In particular, both J (n, 1), the complete graph Kn, and J (n, 2), the strongly regular triangular graph Tn, support fast quantum spatial search. Wong showed that continuous-time quantum walk search on J (n, 3) also supports fast search. The problem is reconsidered in the language of scattering quantum walk, a type of discrete-time quantum walk. Here the search space is confined to a low-dimensional subspace corresponding to the collapsed graph. Using matrix perturbation theory, we show that discrete-time quantum walk search on J (n, 3) also achieves full quantum speedup. The analytical method can also be applied to general Johnson graphs J (n, k) with fixed k.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61802002, 61502101) and the Natural Science Foundation of Anhui Province, China (Grant No. 1708085MF162).
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Xue, Xl., Ruan, Y. & Liu, Zh. Discrete-time quantum walk search on Johnson graphs. Quantum Inf Process 18, 50 (2019). https://doi.org/10.1007/s11128-018-2158-5
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DOI: https://doi.org/10.1007/s11128-018-2158-5