An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree

Andris Ambainis, Aleksandrs Belovs

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.