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：我们考虑的是测试和学习金额(knk$-juntas):美元-quit 单质矩阵问题,该矩阵对美元(qubit)和其余的特性不起作用。作为我们的主要算法结果,我们给出了(a) 美元(sqrt{k}) 美元-query 量算法,可以将美元-juntas与“far”和“k-junta”的单质矩阵区分开来;以及(b) 美元(4k) 美元-query 算法,用于学习美元-juntas。我们在测试美元(kuntas) 和学习金额(kunk) juntas 的上限上加上了近似匹配的 $(sqrt{k}) 美元和 $(mega) 美元(frac{4k}kk}) 的下限。我们的技术是四重分析,并使用qibits对单位的影响概念。