On Testing and Learning Quantum Junta Channels

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 $O(k)$-query algorithm to distinguish whether the given channel is $k$-junta channel or is far from any $k$-junta channels, and a lower bound $\Omega (\sqrt{k})$ on the number of queries and 2) An $\widetilde{O}( 4^{k} )$-query algorithm to learn a $k$-junta channel, and a lower bound $\Omega ( 4^{k}/k )$ on the number of queries. This 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). The distance metric we consider in this paper is obtained by Fourier analysis, which is essentially the L2-distance between Choi representations. Besides, we introduce Influence-Sample to replace Fourier-Sample proposed in(Atici and Servedio, 2007). Our Influence-Sample includes only single-qubit operations and results in only constant-factor decrease in efficiency.