Computing Shor's algorithmic steps with classical light beams

TL;DR

This paper demonstrates that classical light beams with entangled properties can simulate key steps of Shor's quantum factoring algorithm, using classical manipulations and optical interference patterns to identify factors of integers.

Contribution

It introduces a novel classical optical approach to simulate quantum algorithms, specifically implementing Shor's algorithm steps with classical light and interference patterns.

Findings

Classical entangled light beams can mimic quantum parallelism.

Interference patterns encode the multiplicative order for factoring.

Simulation successfully demonstrated for N=15.

Abstract

When considered as orthogonal bases in distinct vector spaces, the unit vectors of polarization directions and the Laguerre-Gaussian modes of polarization amplitude are inseparable, constituting a so-called classical entangled light beam. Equating this classical entanglement to quantum entanglement necessary for computing purpose, we show that the parallelism featured in Shor's factoring algorithm is equivalent to the concurrent light-path propagation of an entangled beam or pulse train. A gedanken experiment is proposed for executing the key algorithmic steps of modular exponentiation and Fourier transform on a target integer $N$ using only classical manipulations on the amplitudes and polarization directions. The multiplicative order associated with the sought-after integer factors is identified through a four-hole diffraction interference from sources obtained from the entangled beam profile. The unique mapping from the fringe patterns to the computed order is demonstrated through simulations for the case $N=15$.