Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

TL;DR

This paper introduces a quantum algorithm for k-junta testing with significantly improved query complexity, leveraging a novel quantum approach to gapped group testing, and establishes a lower bound on query complexity.

Contribution

The paper presents the first quantum algorithm for gapped group testing with a quartic improvement and a new quantum algorithm for junta testing with quadratic query complexity reduction.

Findings

Quantum junta testing query complexity:  O( ext{k}/psilon)

Gapped group testing quantum algorithm with quartic improvement

Lower bound of (k^{1/3}) queries for junta testing

Abstract

In the $k$-junta testing problem, a tester has to efficiently decide whether a given function $f:\{0,1\}^n\rightarrow \{0,1\}$ is a $k$-junta (i.e., depends on at most $k$ of its input bits) or is $\epsilon$-far from any $k$-junta. Our main result is a quantum algorithm for this problem with query complexity $\tilde O(\sqrt{k/\epsilon})$ and time complexity $\tilde O(n\sqrt{k/\epsilon})$. This quadratically improves over the query complexity of the previous best quantum junta tester, due to At\i c\i\ and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of $\Omega(k^{1/3})$ queries for junta-testing (for constant $\epsilon$).