Quantum Phase Estimation Algorithm for Finding Polynomial Roots

TL;DR

This paper presents a quantum algorithm that efficiently finds roots of high-degree polynomials by leveraging quantum eigenvalue estimation on companion matrices, potentially outperforming classical methods for large n.

Contribution

The study develops a quantum algorithm utilizing eigenvalue estimation on quantum circuits for polynomial root finding, improving resource scaling over classical algorithms.

Findings

Quantum circuit for companion matrix constructed

Eigenvalue estimation used to find polynomial roots

Resource scaling is polynomial in degree n

Abstract

Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum algorithm for finding the roots of nth degree polynomials where n is any positive integer. In classical algorithm, the resources required for solving this problem increase drastically when n increases and it would be impossible to practically solve the problem when n is large. It was found that any polynomial can be rearranged into a corresponding companion matrix, whose eigenvalues are roots of the polynomial. This leads to a possibility to perform a quantum algorithm where the number of computational resources increase as a polynomial of n. In this study, we construct a quantum circuit representing the companion matrix and use eigenvalue estimation technique to find roots of polynomial.