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Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

Quantum algorithms and approximating polynomials for composed functions with shared inputs

Abstract

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be a Boolean function and consider a function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q(f), we give an algorithm for evaluating F using quantum queries. This improves on the bound of that follows by treating each conjunction independently, and is tight for worst-case choices of f. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f.
By recursively applying our composition theorems, we obtain a nearly optimal Õ(n1–2–d) upper bound on the quantum query complexity and approximate degree of linear-size depth-d AC0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC0 circuits. We also show that any substantially faster learning algorithm will require fundamentally new techniques.

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cover image Proceedings
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 662 - 678
Editor: Timothy M. Chan, University of Illinois at Urbana-Champaign, USA
ISBN (Online): 978-1-61197-548-2

History

Published online: 2 January 2019

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