A Quantum Algorithm Detecting Concentrated Maps.

Abstract

We consider an arbitrary mapping f: {0, …, N − 1} → {0, …, N − 1} for N = 2n, n some number of quantum bits. Using N calls to a classical oracle evaluating f(x) and an N-bit memory, it is possible to determine whether f(x) is one-to-one. For some radian angle 0 ≤ θ ≤ π/2, we say f(x) is θ − concentrated if and only if for some given ψ0 and any 0 ≤ x ≤ N − 1. We present a quantum algorithm that distinguishes a θ-concentrated f(x) from a one-to-one f(x) in O(1) calls to a quantum oracle function Uf with high probability. For 0 < θ < 0.3301 rad, the quantum algorithm outperforms random (classical) evaluation of the function testing for dispersed values (on average). Maximal outperformance occurs at rad.