Quantum Algorithms for Learning Symmetric Juntas via Adversary Bound

Abstract

In this paper, we study the following variant of the junta learning problem. We are given oracle access to a Boolean function f on n variables that only depends on k variables, and, when restricted to them, equals some predefined function h. The task is to identify the variables the function depends on. This is a generalisation of the Bernstein-Vazirani problem (when h is the XOR function) and the combinatorial group testing problem (when h is the OR function). We analyse the general case using the adversary bound, and give an alternative formulation for the quantum query complexity of this problem. We construct optimal quantum query algorithms for the cases when h is the OR function (complexity is square root of k) or the exact-half function (complexity is the fourth power root of k). The first algorithm resolves an open problem from. For the case when h is the majority function, we prove an upper bound of the fourth power root of k. We obtain a quartic improvement when compared to the randomised complexity (if h is the exact-half or the majority function), and a quadratic one when compared to the non-adaptive quantum complexity (for all functions considered in the paper).