Computing Boolean Functions: Exact Quantum Query Algorithms and Low Degree Polynomials

TL;DR

This paper explores the relationship between quantum query complexity and polynomial degree for Boolean functions, presenting new algorithms and methods to identify functions with significant complexity gaps.

Contribution

It introduces novel quantum algorithms for Boolean functions and methods to construct functions with large gaps between deterministic complexity and polynomial degree.

Findings

Quantum algorithms with lower complexity than deterministic ones.

Methods for constructing Boolean functions with large complexity gaps.

Insights into the relation between polynomial degree and quantum query complexity.

Abstract

In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with quantum query algorithm complexity lower than the deterministic one. Relation between the degree of representing polynomial and potentially possible quantum algorithm complexity has been already described; unfortunately, there are few examples of quantum algorithms to illustrate theoretical evaluation of the complexity. Work in this direction was aimed (1) to construct effective quantum query algorithms for computing Boolean functions, (2) to design methods for Boolean function construction with a large gap between deterministic complexity and degree of representing polynomial. In this paper we present our results in both directions.