Quantum Adversary (Upper) Bound

Abstract

We describe a method for upper bounding the quantum query complexity of certain boolean formula evaluation problems, using fundamental theorems about the general adversary bound. This nonconstructive method gives an upper bound on query complexity without producing an algorithm. For example, we describe an oracle problem that we prove (non-constructively) can be solved in O(1) queries, where the previous best quantum algorithm uses a polynomial number of queries. We then give an explicit O(1) query algorithm based on span programs, and show that for a special case of this problem, there exists a O(1) query algorithm that uses the quantum Haar transform. This special case is a potentially interesting problem in its own right, which we call the Haar Problem.