# Theory of Quantum Path Computing with Fourier Optics and Future Applications for Quantum Supremacy, Neural Networks and Nonlinear Schr\"odinger Equations

## TL;DR

This paper introduces a novel photonic quantum path computing framework using Fourier optics, enabling scalable quantum problem solving without photon interactions, with applications in quantum supremacy, neural networks, and nonlinear Schrödinger equations.

## Contribution

It provides a theoretical model for photonic QPC using Fourier optical setups, extending its problem solving capabilities and discussing future quantum computing applications.

## Key findings

- QPC can sample over 2^100 paths for quantum supremacy.
- Theoretical modeling of QPC with Gaussian and Hermite-Gaussian sources.
- Potential applications include quantum neural networks and nonlinear Schrödinger solutions.

## Abstract

The scalability, error correction and practical problem solving are important challenges for quantum computing (QC) as more emphasized by quantum supremacy (QS) experiments. Quantum path computing (QPC), recently introduced for linear optic based QCs (LOQCs) as an unconventional design, targets to obtain scalability and practical problem solving. It samples the intensity from the interference of exponentially increasing number of propagation paths obtained in multi-plane diffraction (MPD) of classical particle sources. QPC exploits MPD based quantum temporal correlations of the paths and freely entangled projections a<t different time instants, for the first time, with the classical light source and intensity measurement while not requiring photon interactions or single photon sources and receivers. In this article, photonic QPC is defined, theoretically modeled and numerically analyzed for arbitrary Fourier optical or quadratic phase set-ups while utilizing both Gaussian and Hermite-Gaussian source laser modes. Problem solving capabilities already including partial sum of Riemann theta functions are extended. Important future applications, implementation challenges and open issues such as universal computation and quantum circuit implementations determining the scope of QC capabilities are discussed. The applications include QS experiments reaching more than $2^{100}$ Feynman paths, quantum neuron implementations and solutions of nonlinear Schr\"odinger equation.