Average-case quantum query complexity
Andris Ambainis, Ronald de Wolf · Journal of Physics A Mathematical and General · 2001
We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than the classical complexity. A preliminary version of this paper appeared in the Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science (STACS'2000) (Lecture Notes in Computer Science vol 1770) (Berlin: Springer).