Quantum speedup to some types of polynomial equations

TL;DR

This paper develops quantum algorithms for solving specific polynomial equations, including linear divisibility, quadratic congruences, and exponential congruences in finite fields, demonstrating quantum speedups for these problems.

Contribution

It introduces new quantum algorithms for polynomial equations, extending previous work and providing simpler solutions for exponential congruences in finite fields.

Findings

Quantum algorithms for linear divisibility and quadratic congruences

Simplified quantum solution for exponential congruence equations

Generalizations of quantum algorithms for polynomial equations

Abstract

In this paper, we consider three types of polynomial equations in quantum computer: linear divisibility equation, which belongs to a special type of binary-quadratic Diophantine equation; quadratic congruence equation with restriction in the solution and exponential congruence equation in finite field. Quantum algorithms based on Grover's algorithm and Shor's algorithm to these problems are given. As for the exponential congruence equation, which has been considered by Dam and Shparlinski \cite{dam} at 2008, a relatively simple quantum algorithm is given here. And some other results and generalizations are discovered.