On the various ways of quantum implementation of the modular exponentiation function for Shor’s factorization

Abstract

The content of this paper is a detailed analysis of possible ways how to quantum implement a key part of Shor’s factorization algorithm, the modular exponentiation function. This implementation is a bottleneck for performing quantum factorization with polynomial complexity, which would make it possible to factorize really large numbers in a reasonable amount of time. In this paper, not only the general theory is presented, but also the results of successful factorizations of the numbers 247 and 143 using Shor’s algorithm from a quantum computer simulator. An interesting fact is that no ancillary qubits were needed in these factorizations. Based on the content of the paper, the conclusion also suggests possible future work on the development of this modular exponentiation function implementation.