Polynomial degree vs. quantum query complexity

Abstract

The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f . This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with polynomial degree M and quantum query complexity Ω ( M 1.321 … ) . This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a generalized version of the quantum adversary method.