An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree

TL;DR

This paper demonstrates an exponential separation between quantum query complexity and polynomial degree for a partial Boolean function, highlighting a significant gap in their relationship for such functions.

Contribution

It provides the first known exponential separation between exact polynomial degree and approximate quantum query complexity for partial functions.

Findings

Exponential separation between quantum query complexity and polynomial degree.

Constant versus polynomial separation with unbounded alphabet size.

Advances understanding of quantum versus classical complexity measures.

Abstract

While it is known that there is at most a polynomial separation between quantum query complexity and the polynomial degree for total functions, the precise relationship between the two is not clear for partial functions. In this paper, we demonstrate an exponential separation between exact polynomial degree and approximate quantum query complexity for a partial Boolean function. For an unbounded alphabet size, we have a constant versus polynomial separation.