Quantum conditional query complexity

TL;DR

This paper introduces quantum conditional oracles and demonstrates their advantages in distribution testing, boolean function analysis, and quantum state property testing, achieving improved quantum speed-ups over existing methods.

Contribution

It defines quantum conditional oracles and explores their applications, providing new algorithms with quantum speed-ups for distribution and state property testing.

Findings

Speed-ups in distribution testing algorithms

Efficient quantum algorithms for boolean function properties

Sub-linear quantum algorithm for quantum state testing

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 speed-ups over the best known 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 $\epsilon$-far from balanced; and (c) give a sub-linear algorithm, requiring $\tilde{O}(n^{3/4}/\epsilon)$ queries, for testing whether an $n$-dimensional quantum state is maximally mixed or not.