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.
Discrete-time quantum walks Quantum spatial search Johnson graphs
This is a preview of subscription content, log in to check access.
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).