The quantum query complexity of learning multilinear polynomials

Authors:
Ashley Montanaro
Affiliations:
Centre for Quantum Information and Foundations, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK
Venue:
Information Processing Letters
Year:
2012

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Abstract

In this note we study the number of quantum queries required to identify an unknown multilinear polynomial of degree d in n variables over a finite field F"q. Any bounded-error classical algorithm for this task requires @W(n^d) queries to the polynomial. We give an exact quantum algorithm that uses O(n^d^-^1) queries for constant d, which is optimal. In the case q=2, this gives a quantum algorithm that uses O(n^d^-^1) queries to identify a codeword picked from the binary Reed-Muller code of order d.