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A note on Simon's period-finding quantum algorithm

Yang Li, Hongwei Li

Published Jun 9, 2013• 0 citations
OpenAlex

Abstract

It is believed that there is no efficient classical algorithm to determine the linear structure of Boolean function. We investigate an extension of Simon's period-finding quantum algorithm, and propose an efficient quantum algorithm to determine the linear structure of Boolean function.

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A quantum algorithm to determine approximations of linear structures of Boolean functions is presented and analysed. Similar results have already been published (see Simon's algorithm) but only for some promise versions of the problem, and it has been shown that no exponential quantum speedup can be obtained for the general (no promise) version of the problem. In this paper, no additional promise assumptions are made. The approach presented is based on the method used in the Bernstein–Vazirani algorithm to identify linear Boolean functions and on ideas from Simon's period finding algorithm. A proper combination of these two approaches results here to a polynomial-time approximation to the linear structures set. Specifically, we show how the accuracy of the approximation with high probability changes according to the running time of the algorithm. Moreover, we show that the time required for the linear structure determine problem with high success probability is related to so called relative differential uniformityδfof a Boolean functionf. Smaller differential uniformity is, shorter time is needed.