On Testing and Learning Quantum Junta Channels

TL;DR

This paper introduces algorithms for testing and learning quantum $k$-junta channels, establishing query complexity bounds, and develops a Fourier analysis framework over superoperators to advance quantum property testing.

Contribution

It provides the first known algorithms and bounds for quantum junta channel testing and learning, and extends Fourier analysis techniques to superoperator spaces.

Findings

O(k) query algorithm for testing junta channels

ig ddd lower bound ddddd on queries

ig ddddd query algorithm for learning junta channels

Abstract

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.