Uselessness for an Oracle Model with Internal Randomness

TL;DR

This paper explores a generalized oracle model with internal randomness, demonstrating quantum advantages in certain problems and establishing bounds on classical versus quantum query complexities.

Contribution

It introduces a generalized oracle model with internal randomness, showing quantum advantages and providing bounds on classical and quantum query complexities.

Findings

Quantum algorithms solve certain problems classically impossible with a single query.

Classical and quantum query complexities are related by a factor of two in the internal randomness model.

Conditions are provided to determine when oracle problems cannot surpass random guessing.

Abstract

We consider a generalization of the standard oracle model in which the oracle acts on the target with a permutation selected according to internal random coins. We describe several problems that are impossible to solve classically but can be solved by a quantum algorithm using a single query; we show that such infinity-vs-one separations between classical and quantum query complexities can be constructed from much weaker separations. We also give conditions to determine when oracle problems---either in the standard model, or in any of the generalizations we consider---cannot be solved with success probability better than random guessing would achieve. In the oracle model with internal randomness where the goal is to gain any nonzero advantage over guessing, we prove (roughly speaking) that $k$ quantum queries are equivalent in power to $2k$ classical queries, thus extending results of Meyer and Pommersheim.