Quantum Algorithms for Learning Symmetric Juntas via the 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. When $h$ is the XOR or the OR function, this gives a restricted variant of the Bernstein-Vazirani or the combinatorial group testing problem, respectively. 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 $\Theta (\sqrt{k})$) or the exact-half function (complexity is $\Theta (k^{1/4})$). The first algorithm resolves an open problem from arXiv:1210.1148. For the case when $h$ is the majority function, we prove an upper bound of $O(k^{1/4})$. All these algorithms can be made exact. 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).

19 pages, many parts re-written, some references and minor results added

References in corpus (6)

• Exponential algorithmic speedup by quantum walk
• Improved Bounds on Quantum Learning Algorithms
• Learning-Graph-Based Quantum Algorithm for k-distinctness
• An optimal quantum algorithm for the oracle identification problem
• Easy and hard functions for the Boolean hidden shift problem
• All Quantum Adversary Methods are Equivalent

Cited by in corpus (3)

• Quantum algorithms: an overview
• A Survey of Quantum Property Testing
• Quantum Algorithm for Monotonicity Testing on the Hypercube