How Low Can Approximate Degree and Quantum Query Complexity be for Total Boolean Functions?

TL;DR

This paper establishes a tight lower bound of Omega(log n / loglog n) for the approximate degree and bounded-error quantum query complexity of total Boolean functions, and provides algorithms that meet this bound.

Contribution

It proves the optimal lower bound for approximate degree and quantum query complexity, and constructs functions that achieve this bound.

Findings

Lower bound of Omega(log n / loglog n) for approximate degree and quantum query complexity.

Existence of functions that meet this lower bound with quantum algorithms.

Abstract

It has long been known that any Boolean function that depends on n input variables has both degree and exact quantum query complexity of Omega(log n), and that this bound is achieved for some functions. In this paper we study the case of approximate degree and bounded-error quantum query complexity. We show that for these measures the correct lower bound is Omega(log n / loglog n), and we exhibit quantum algorithms for two functions where this bound is achieved.