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? We present the first example of a total Boolean function $f(x_1,\ldots, x_N)$ 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 $O(N^{0.8675\ldots})$ 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 $O(N^{0.8675\ldots}\log N)$ quantum bits but requires $\Omega(N)$ bits of communication for classical protocols.