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computational complexity

, Volume 24, Issue 2, pp 255–293 | Cite as

Quantum Algorithms for Learning Symmetric Juntas via the Adversary Bound

  • Aleksandrs BelovsEmail author
Article

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 analyze 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 Ambainis & Montanaro (Quantum Inf Comput 14(5&6): 439–453, 2014). 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 randomized 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).

Keywords

Quantum query algorithms computational learning theory combinatorial group testing representation theory of the symmetric group semi-definite optimization 

Subject classification

68Q12 81P68 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Computer Science and Artificial, Intelligence LaboratoryMassachusetts Institute of TechnologyCambridgeUSA

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