Quantum conditional query complexity

Abstract

We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain highly efficient quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an n-input m-output boolean function is balanced or ϵ-far from balanced; and (c) give an algorithm, requiring Õ(n/ϵ) queries, for testing whether an n-dimensional quantum state is maximally mixed or not.