01395nas a2200169 4500008004100000245006500041210006500106260001500171300001000186490000700196520090400203100002301107700001901130700001701149700002201166856003701188 2013 eng d00aEasy and hard functions for the Boolean hidden shift problem0 aEasy and hard functions for the Boolean hidden shift problem c2013/04/16 a50-790 v223 a We study the quantum query complexity of the Boolean hidden shift problem.
Given oracle access to f(x+s) for a known Boolean function f, the task is to
determine the n-bit string s. The quantum query complexity of this problem
depends strongly on f. We demonstrate that the easiest instances of this
problem correspond to bent functions, in the sense that an exact one-query
algorithm exists if and only if the function is bent. We partially characterize
the hardest instances, which include delta functions. Moreover, we show that
the problem is easy for random functions, since two queries suffice. Our
algorithm for random functions is based on performing the pretty good
measurement on several copies of a certain state; its analysis relies on the
Fourier transform. We also use this approach to improve the quantum rejection
sampling approach to the Boolean hidden shift problem.
1 aChilds, Andrew, M.1 aKothari, Robin1 aOzols, Maris1 aRoetteler, Martin uhttp://arxiv.org/abs/1304.4642v1