Toward an Optimal Quantum Algorithm for Polynomial Factorization over Finite Fields

TL;DR

This paper introduces a randomized quantum algorithm that significantly improves polynomial factorization over finite fields, breaking classical complexity barriers with an expected near-linear time complexity.

Contribution

It presents the first quantum algorithm with sub-quadratic complexity for polynomial factorization over finite fields, surpassing classical methods and breaking the 3/2-exponent barrier.

Findings

Expected complexity is near-linear in polynomial degree and logarithmic in field size.

Higher complexity for a negligible subset of polynomials, maintaining overall efficiency.

Breaks the classical 3/2-exponent barrier for polynomial factorization.

Abstract

We present a randomized quantum algorithm for polynomial factorization over finite fields. For polynomials of degree $n$ over a finite field $\F_q$, the average-case complexity of our algorithm is an expected $O(n^{1 + o(1)} \log^{2 + o(1)}q)$ bit operations. Only for a negligible subset of polynomials of degree $n$ our algorithm has a higher complexity of $O(n^{4 / 3 + o(1)} \log^{2 + o(1)}q)$ bit operations. This breaks the classical $3/2$-exponent barrier for polynomial factorization over finite fields \cite{guo2016alg}.