Superlinear advantage for exact quantum algorithms

TL;DR

This paper demonstrates the first example of a Boolean function where exact quantum algorithms outperform classical deterministic algorithms with a superlinear advantage, requiring significantly fewer queries.

Contribution

It introduces a new Boolean function for which exact quantum algorithms have a superlinear query complexity advantage over classical algorithms.

Findings

Quantum algorithms can compute the new function with O(N^{0.8675...}) queries.

Classical algorithms require N queries for the same function.

First known superlinear advantage for exact quantum algorithms over deterministic ones.

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 inputs 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(x_1, ..., x_N) for which exact quantum algorithms have superlinear advantage over the 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...}) queries.