The quantum query complexity of learning multilinear polynomials

TL;DR

This paper investigates the quantum query complexity for learning multilinear polynomials over finite fields, providing an optimal quantum algorithm that significantly reduces the number of queries needed compared to classical methods.

Contribution

The paper introduces an exact quantum algorithm that identifies multilinear polynomials with fewer queries, achieving optimal query complexity for constant degree d.

Findings

Quantum algorithm uses O(n^(d-1)) queries for constant d

Classical algorithms require Omega(n^d) queries

Optimal query complexity achieved for learning multilinear polynomials

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 Omega(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.