Quantum algorithm for the root-finding problem

Vicente Moret‐Bonillo

Sun Guo-don

Root finding problem is a hard problem in computational number theory. To improve the efficiency of solving the problem and expand the scope of applications of quantum computing,the quantum algorithms for solving the root finding problem are analyzed in this paper. Based on the two important quantum algorithms—Shor's algorithm and Grover's algorithm, two quantum mechanical algorithms are proposed to solve the root finding problem,which are named RF-Shor algorithm and RFGrover algorithm. According to the analysis,the RF-Shor algorithm can find all the solutions to the root finding problem with a probability closed to 100%,with a consumption of polynomial quantum gates; the RF-Grover algorithm does not involve any classical method to improve the search efficiency and one of the k solutions to the root finding problem can be obtained in O(M/k) steps with a probability of at least50%.

Koji Nagata, Tadao Nakamura, Han J.F. Geurdes +4 more

Guodong Sun, Shenghui Su, Maozhi Xu

Quantum computation is a new computing model based on fundamental quantum mechanical principle. Grover's algorithm finds the solution for a searching problem in the square root time of exhaustive search. Brassard, Hoyer, Tapp's algorithm counts the number of solutions for a searching problem. Through exploiting the two quantum algorithms, we propose a quantum algorithm for solving a new cryptography problem -- polynomial root finding problem, which could be used to design a cryptosystem. The algorithm will take O(rootM/t) steps for finding one of the t solutions to the problem, where M is the modular of the equation. The success rate of the algorithm is a constant and the cost of the algorithm depends on the calculations of modular exponentiation and the number of iterations.

Weng-Long Chang, Athanasios V. Vasilakos

Theerapat Tansuwannont, Surachate Limkumnerd, Sujin Suwanna +1 more

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.