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 $\widetilde{O}\left(\sqrt{k}\right)$-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 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).

翻译：我们研究量子$k$-Junta信道的测试与学习问题，这类信道是作用在$n$量子比特上、但仅对其中至多$k$个量子比特产生非平凡作用，其余量子比特保持不变的$n$量子比特到$n$量子比特的量子信道。我们得到以下结果：1. 一个$\widetilde{O}\left(\sqrt{k}\right)$次查询的算法，用于区分给定信道是否为$k$-Junta信道或远离任何$k$-Junta信道，以及查询次数的下界$\Omega\left(\sqrt{k}\right)$；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，以及查询次数的下界$\Omega\left(4^k/k\right)$。这解决了Chen等人(2023)提出的一个开放问题。为解决这些问题，我们建立了超算子空间上的傅里叶分析框架，并证明了若干基本性质，该框架将Montanaro和Osborne(2010)引入的算子空间上的傅里叶分析进行了推广。