Testing and Learning Quantum Juntas Nearly Optimally

TL;DR

This paper develops nearly optimal quantum algorithms for testing and learning quantum $k$-juntas, which are $n$-qubit unitaries acting non-trivially on only $k$ qubits, using Fourier analysis and influence measures.

Contribution

It introduces quantum algorithms with query complexities close to theoretical lower bounds for testing and learning quantum $k$-juntas.

Findings

Quantum testing algorithm distinguishes $k$-juntas with $ ilde{O}(\sqrt{k})$ queries.

Quantum learning algorithm learns $k$-juntas with $O(4^k)$ queries.

Lower bounds match the upper bounds up to logarithmic factors.

Abstract

We consider the problem of testing and learning quantum $k$-juntas: $n$-qubit unitary matrices which act non-trivially on just $k$ of the $n$ qubits and as the identity on the rest. As our main algorithmic results, we give (a) a $\widetilde{O}(\sqrt{k})$-query quantum algorithm that can distinguish quantum $k$-juntas from unitary matrices that are "far" from every quantum $k$-junta; and (b) a $O(4^k)$-query algorithm to learn quantum $k$-juntas. We complement our upper bounds for testing quantum $k$-juntas and learning quantum $k$-juntas with near-matching lower bounds of $\Omega(\sqrt{k})$ and $\Omega(\frac{4^k}{k})$, respectively. Our techniques are Fourier-analytic and make use of a notion of influence of qubits on unitaries.