Remarks on Quantum Modular Exponentiation and Some Experimental Demonstrations of Shor's Algorithm

TL;DR

This paper critiques existing methods for quantum modular exponentiation in Shor's algorithm, highlights flaws in prior descriptions, and clarifies conditions for experimental demonstrations to be valid.

Contribution

It identifies inaccuracies in previous descriptions of quantum modular exponentiation and emphasizes the correct conditions for experimental demonstrations of Shor's algorithm.

Findings

Existing descriptions of quantum modular exponentiation are flawed.

Many experimental demonstrations violate necessary conditions for validity.

Clarifies the correct parameter choices for experimental implementations.

Abstract

An efficient quantum modular exponentiation method is indispensible for Shor's factoring algorithm. But we find that all descriptions presented by Shor, Nielsen and Chuang, Markov and Saeedi, et al., are flawed. We also remark that some experimental demonstrations of Shor's algorithm are misleading, because they violate the necessary condition that the selected number $q=2^s$, where $s$ is the number of qubits used in the first register, must satisfy $n^2 \leq q < 2n^2$, where $n$ is the large number to be factored.