We study search by quantum walk on a two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [AKR05]. We show what the most natural coin transformation — Grover’s diffusion transformation — has a wide class of exceptional configurations of marked locations, for which the probability of finding any of the marked locations does not grow over time. This extends the class of known exceptional configurations; until now the only known such configuration was the “diagonal construction” by [AR08].

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Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of SODA 2005, pp. 1099–1108 (2005)Google Scholar

[AR08]

Ambainis, A., Rivosh, A.: Quantum walks with multiple or moving marked locations. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 485–496. Springer, Heidelberg (2008)CrossRefGoogle Scholar

[BS06]

Buhrman, H., Spalek, R.: Quantum verification of matrix products. In: Proceedings of SODA 2006, pp. 880–889 (2006)Google Scholar

[CC+03]

Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by a quantum walk. In: Proceedings of 35th ACM STOC, pp. 59–68 (2003)Google Scholar

[MSS05]

Magniez, F., Santha, M., Szegedy, M.: An \(O(n^{1.3})\) quantum algorithm for the triangle problem. In: Proceedings of SODA 2005, pp. 413–424 (2005)Google Scholar

[NR15]

Nahimovs, N., Rivosh, A.: Quantum walks on two-dimensional grids with multiple marked locations. arXiv:1507.03788 (2015)