Simon's problem for linear functions

10/29/2018

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by   Joran van Apeldoorn, et al.

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Simon's problem asks the following: determine if a function f: {0,1}^n →{0,1}^n is one-to-one or if there exists a unique s ∈{0,1}^n such that f(x) = f(x ⊕ s) for all x ∈{0,1}^n, given the promise that exactly one of the two holds. A classical algorithm that can solve this problem for every f requires 2^Ω(n) queries to f. Simon showed that there is a quantum algorithm that can solve this promise problem for every f using only O(n) quantum queries to f. A matching lower bound on the number of quantum queries was given by Koiran et al., even for functions f: F_p^n→F_p^n. We give a short proof that O(n) quantum queries is optimal even when we are additionally promised that f is linear. This is somewhat surprising because for linear functions there even exists a classical n-query algorithm.