Testing and Learning Quantum Juntas Nearly Optimally

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 $\tilde{O}(\sqrt{k})$-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 $\mathrm{\Omega }(\sqrt{k})$ and $\mathrm{\Omega }(\frac{4^k}{k})$, respectively. Our techniques are Fourier-analytic and make use of a notion of influence of qubits on unitaries.