We consider the problem of testing and learning quantum k-juntas: n-qubit unitary matrices which act non-trivially on just k of the n qubits and as the identity on the rest. As our main algorithmic results, we give 1. A -query quantum algorithm that can distinguish quantum k-juntas from unitary matrices that are “far” from every quantum k-junta; and 2. A O(4k)-query algorithm to learn quantum k-juntas. We complement our upper bounds for testing and learning quantum k-juntas with near-matching lower bounds of and Ω(4k/k), respectively. Our techniques are Fourier-analytic and make use of a notion of influence of qubits on unitaries. * The full version of the paper can be accessed at https://arxiv.org/abs/2207.05898
Given a black-box representing an unknown Boolean function
f of
n variables, in this paper we propose a fast quantum algorithm to test whether or not a certain variable in the function
f is a junta variable. The proposed algorithm creates entanglement between the variable under test and an auxiliary qubit, where the entanglement is measured using concurrence measure to decide if the variable is junta. The paper shows applications to the proposed algorithm in learning and categorization of Boolean functions.