Unbounded Error Quantum Query Complexity

TL;DR

This paper demonstrates that in unbounded error quantum query complexity, quantum algorithms are exactly half as complex as classical ones, and explores the weakly unbounded error setting revealing significant quantum advantages.

Contribution

It establishes the precise relationship between quantum and classical unbounded error query complexities and analyzes the weakly unbounded error model for Boolean functions.

Findings

Quantum unbounded error query complexity is half of classical for all Boolean functions.

Optimality of the Buhrman-Cleve-Wigderson conversion in the unbounded error setting.

Logarithmic separation between quantum and classical complexities in the weakly unbounded error model.

Abstract

This work studies the quantum query complexity of Boolean functions in a scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, we show that the "black-box" approach to convert quantum query algorithms into communication protocols by Buhrman-Cleve-Wigderson [STOC'98] is optimal even in the unbounded error setting. We also study a setting related to the unbounded error model, called the weakly unbounded error setting, where the cost of a query algorithm is given by q+log(1/2(p-1/2)), where q is the number of queries made and p>1/2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight Theta(log n) separation between quantum and classical query complexity in the weakly unbounded error setting for a partial Boolean function. We also show the asymptotic equivalence between them for some well-studied total Boolean functions.