Simon's problem for linear functions

Abstract

Simon's problem asks the following: determine if a function $f: \{0,1\}^n \rightarrow \{0,1\}^n$ is one-to-one or if there exists a unique $s \in \{0,1\}^n$ such that $f(x) = f(x \oplus s)$ for all $x \in \{0,1\}^n$, given the promise that exactly one of the two holds. A classical algorithm that can solve this problem for every $f$ requires $2^{\Omega(n)}$ queries to $f$. Simon showed that there is a quantum algorithm that can solve this promise problem for every $f$ using only $\mathcal O(n)$ quantum queries to $f$. A matching lower bound on the number of quantum queries was given by Koiran et al., even for functions $f: {\mathbb{F}_p^n} \to {\mathbb{F}_p^n}$. We give a short proof that $\mathcal O(n)$ 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.