We consider the problems of testing and learning quantum $k$-junta channels, which are $n$-qubit to $n$-qubit quantum channels acting non-trivially on at most $k$ out of $n$ qubits and leaving the rest of qubits unchanged. We show the following. 1. An $O\left(k\right)$-query algorithm to distinguish whether the given channel is $k$-junta channel or is far from any $k$-junta channels, and a lower bound $\Omega\left(\sqrt{k}\right)$ on the number of queries; 2. An $\widetilde{O}\left(4^k\right)$-query algorithm to learn a $k$-junta channel, and a lower bound $\Omega\left(4^k/k\right)$ on the number of queries. This gives the first junta channel testing and learning results, and partially answers an open problem raised by Chen et al. (2023). In order to settle these problems, we develop a Fourier analysis framework over the space of superoperators and prove several fundamental properties, which extends the Fourier analysis over the space of operators introduced in Montanaro and Osborne (2010). Besides, we introduce $\textit{Influence-Sample}$ to replace $\textit{Fourier-Sample}$ proposed in Atici and Servedio (2007). Our $\textit{Influence-Sample}$ includes only single-qubit operations and results in only constant-factor decrease in efficiency.

翻译：我们考虑量子$k$-Junta信道的测试与学习问题，这类信道是$n$量子比特到$n$量子比特的量子信道，其非平凡作用仅涉及$n$个量子比特中的至多$k$个，而其余量子比特保持不变。我们得到以下结果：1. 一个$O\left(k\right)$次查询的算法，用于判断给定信道是否为$k$-Junta信道或远离任何$k$-Junta信道，并给出查询次数的下界$\Omega\left(\sqrt{k}\right)$；2. 一个$\widetilde{O}\left(4^k\right)$次查询的算法，用于学习一个$k$-Junta信道，并给出查询次数的下界$\Omega\left(4^k/k\right)$。这首次给出了Junta信道测试与学习的结果，并部分回答了Chen等人（2023）提出的开放问题。为解决这些问题，我们在超算子空间上建立了傅里叶分析框架，并证明了若干基本性质，这扩展了Montanaro和Osborne（2010）在算子空间上引入的傅里叶分析。此外，我们引入$\textit{Influence-Sample}$以替代Atici和Servedio（2007）提出的$\textit{Fourier-Sample}$。我们的$\textit{Influence-Sample}$仅包含单量子比特操作，且仅导致效率上的常数因子降低。