A quantum algorithm for finding hidden periodicities — exponential speedup over classical
Given a black-box function f : {0,1}ⁿ → {0,1}ⁿ with the promise that there exists a secret string s such that f(x) = f(y) if and only if x ⊕ y ∈ {0ⁿ, s}, find s. Classically this requires O(2n/2) queries; Simon's quantum algorithm solves it in O(n) queries.