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.

翻译：我们研究量子$k$-junta的测试与学习问题：即$n$量子比特幺正矩阵，其仅在$k$个量子比特上非平凡作用，其余量子比特上为单位矩阵。作为主要算法结果，我们给出：(a) 一个$\widetilde{O}(\sqrt{k})$次查询的量子算法，可区分量子$k$-junta与所有量子$k$-junta均"远离"的幺正矩阵；(b) 一个$O(4^k)$次查询的学习量子$k$-junta算法。我们通过$\Omega(\sqrt{k})$和$\Omega(\frac{4^k}{k})$近乎匹配的下界，分别补充了量子$k$-junta测试与学习问题的上界。我们的技术基于傅里叶分析，并利用了量子比特对幺正矩阵影响的概念。