A Continuous Variable Shor Algorithm

TL;DR

This paper develops a continuous variable quantum algorithm inspired by Shor's factoring algorithm, aiming to clarify its structure and provide insights into related quantum algorithms, though efficiency and implementation remain open questions.

Contribution

It introduces a continuous variable analogue of Shor's algorithm, enhancing understanding of quantum factoring and hidden subgroup algorithms in a continuous setting.

Findings

Provides a continuous quantum algorithm for period finding

Offers insights into the structure of Shor's and Hallgren's algorithms

Raises open questions about efficiency and implementability

Abstract

In this paper, we use the methods found in quant-ph/0201095 to create a continuous variable analogue of Shor's quantum factoring algorithm. By this we mean a quantum hidden subgroup algorithm that finds the period P of a function F:R-->R from the reals R to the reals R, where F belongs to a very general class of functions, called the class of admissible functions. One objective in creating this continuous variable quantum algorithm was to make the structure of Shor's factoring algorithm more mathematically transparent, and thereby give some insight into the inner workings of Shor's original algorithm. This continuous quantum algorithm also gives some insight into the inner workings of Hallgren's Pell's equation algorithm. Two key questions remain unanswered. Is this quantum algorithm more efficient than its classical continuous variable counterpart? Is this quantum algorithm or some approximation of it implementable?