Efficient quantum processing of ideals in finite rings

TL;DR

This paper presents a quantum algorithm for efficiently finding a basis for ideals in finite rings and demonstrates its utility in solving various complex ring-theoretic problems rapidly, which are classically hard.

Contribution

It introduces a quantum algorithm that computes ideal bases in polylogarithmic time and applies it to solve multiple fundamental problems in finite ring theory.

Findings

Efficient quantum basis computation for ideals in finite rings.

Quantum algorithms for ideal comparison, intersection, and quotient.

Rapid solutions to problems like element membership, units, inverses, and ring homomorphisms.

Abstract

Suppose we are given black-box access to a finite ring R, and a list of generators for an ideal I in R. We show how to find an additive basis representation for I in poly(log |R|) time. This generalizes a quantum algorithm of Arvind et al. which finds a basis representation for R itself. We then show that our algorithm is a useful primitive allowing quantum computers to rapidly solve a wide variety of problems regarding finite rings. In particular we show how to test whether two ideals are identical, find their intersection, find their quotient, prove whether a given ring element belongs to a given ideal, prove whether a given element is a unit, and if so find its inverse, find the additive and multiplicative identities, compute the order of an ideal, solve linear equations over rings, decide whether an ideal is maximal, find annihilators, and test the injectivity and surjectivity of ring homomorphisms. These problems appear to be hard classically.