The Category of Linear Optical Quantum Computing

TL;DR

This paper reviews the theoretical framework of linear optical quantum computing, focusing on the categorical and diagrammatic representations of photonic circuits, and demonstrates key algorithms like boson sampling within this formalism.

Contribution

It introduces a diagrammatic language for the category of linear optical quantum computing, facilitating understanding and development of photonic algorithms.

Findings

Formalized the category of unitary matrices on bosonic Fock space

Developed a diagrammatic language for photonic algorithms

Demonstrated boson sampling and KLM protocol within the formalism

Abstract

This note reviews the model of computation generated by photonic circuits, comprising edges that are traversed by photons in a single time-bin and vertices given by idealised lossless beam splitters and phase shifters. The circuit model is abstracted as a representation of the symmetric monoidal category of unitary matrices on the bosonic Fock space of multimode photons. A diagrammatic language, designed to aid with the understanding and development of photonic algorithms, is presented that encapsulates the category properties of this representation. As demonstrations of the formalism, the boson sampling scheme and the protocol of Knill, Laflamme and Milburn are developed on the single-rail photonic computer, and a parity-based model for the qudit is investigated.