Quantum algorithms for search with wildcards and combinatorial group testing

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 $O(k)$ queries to solve this problem, as compared with the classical lower bound of $\Omega(k \log(n/k))$ queries.