Quantum algorithms for search with wildcards and combinatorial group testing

TL;DR

This paper presents nearly optimal quantum algorithms for two combinatorial problems: search with wildcards and group testing, significantly reducing query complexity compared to classical methods.

Contribution

It introduces a quantum algorithm for search with wildcards based on state discrimination, and a simple quantum approach for group testing, both outperforming classical bounds.

Findings

Quantum search with wildcards uses O(√n log n) queries, beating classical Omega(n)

Quantum group testing uses O(k log k) queries, better than classical Omega(k log(n/k))

Algorithms are nearly optimal and demonstrate quantum advantage in combinatorial problems.

Abstract

We consider two combinatorial problems. The first we call "search with wildcards": given an unknown n-bit string x, and the ability to check whether any subset of the bits of x is equal to a provided query string, the goal is to output x. We give a nearly optimal O(sqrt(n) log n) quantum query algorithm for search with wildcards, beating the classical lower bound of Omega(n) queries. Rather than using amplitude amplification or a quantum walk, our algorithm is ultimately based on the solution to a state discrimination problem. The second problem we consider is combinatorial group testing, which is the task of identifying a subset of at most k special items out of a set of n items, given the ability to make queries of the form "does the set S contain any special items?" for any subset S of the n items. We give a simple quantum algorithm which uses O(k log k) queries to solve this problem, as compared with the classical lower bound of Omega(k log(n/k)) queries.