# Factoring an integer with three oscillators and a qubit

## TL;DR

This paper proposes a new quantum factoring algorithm that uses just one qubit and three oscillators, regardless of the size of the integer being factored.

## Contribution

The novel contribution is a polynomial-time quantum factoring algorithm requiring only one qubit and three oscillators, independent of the integer size.

## Key findings

- A hybrid qubit-oscillator system can perform continuous variable Fourier transforms and arithmetic operations natively.
- The proposed algorithm uses only one qubit and three oscillators for factoring integers of any size.
- The approach incurs an extensive energy cost but reduces the number of qubits needed compared to Shor’s algorithm.

## Abstract

A common starting point of traditional quantum algorithm design is the notion of a universal quantum computer with a scalable number of qubits. This convenient abstraction mirrors classical computations manipulating bits. It allows for a device-independent development of algorithmic primitives. Here we argue that an alternative approach centered on the physical setup can yield great benefits. As an example, we consider hybrid qubit-oscillator systems with linear optics operations augmented by certain qubit-controlled Gaussian unitaries. The continuous variable Fourier transform and certain arithmetic operations have native realizations in such systems. We put this to algorithmic use and give a polynomial-time quantum factoring algorithm which uses only one qubit and three oscillators, independent of the number being factored.

The number of qubits needed for Shor’s factoring algorithm scales with the size of the integer to be factored. Here, the authors propose an alternative approach that would require only three harmonic oscillators and a qubit, regardless of the size of the integer, but incurring in an extensive energy cost.