Superlinear advantage for exact quantum algorithms

Abstract

A quantum algorithm is exact if, on any input data, it outputs the correct answer with certainty (probability 1). A key question is: how big is the advantage of exact quantum algorithms over their classical counterparts: deterministic algorithms. For total Boolean functions in the query model, the biggest known gap was just a factor of 2: PARITY of N input bits requires N queries classically but can be computed with N/2 queries by an exact quantum algorithm. We present the first example of a Boolean function f(x1, ..., xN) for which exact quantum algorithms have superlinear advantage over deterministic algorithms. Any deterministic algorithm that computes our function must use N queries but an exact quantum algorithm can compute it with O(N0.8675...) queries. A modification of our function gives a similar result for communication complexity: there is a function f which can be computed by an exact quantum protocol that communicates O(N^{0.8675...}) quantum bits but requires Omega(N) bits of communication for classical protocols.