Tolerant Quantum Junta Testing

Junta testing for Boolean functions has sparked a long line of work over recent decades in theoretical computer science, and recently has also been studied for unitary operators in quantum computing. Tolerant junta testing is more general and challenging than the standard version. While optimal tolerant junta testers have been obtained for Boolean functions, there has been no knowledge about tolerant junta testers for unitary operators, which was thus left as an open problem in [Chen, Nadimpalli, and Yuen, SODA2023]. In this paper, we settle this problem by presenting the first algorithm to decide whether a unitary is $\epsilon_1$-close to some quantum $k$-junta or is $\epsilon_2$-far from any quantum $k$-junta, where an $n$-qubit unitary $U$ is called a quantum $k$-junta if it only non-trivially acts on just $k$ of the $n$ qubits. More specifically, we present a tolerant tester with $\epsilon_1 = \frac{\sqrt{\rho}}{8} \epsilon$, $\epsilon_2 = \epsilon$, and $\rho \in (0,1)$, and the query complexity is $O\left(\frac{k \log k}{\epsilon^2 \rho (1-\rho)^k}\right)$, which demonstrates a trade-off between the amount of tolerance and the query complexity. Note that our algorithm is non-adaptive which is preferred over its adaptive counterparts, due to its simpler as well as highly parallelizable nature. At the same time, our algorithm does not need access to $U^\dagger$, whereas this is usually required in the literature.