Quantum Speedup and Categorical Distributivity

TL;DR

This paper explores the categorical foundations of Shor's quantum algorithm, demonstrating that distributivity principles can establish efficiency and equivalence of quantum circuits in various categorical frameworks.

Contribution

It provides a minimal categorical framework for key quantum operations, generalizing the efficiency of Shor's algorithm beyond Hilbert spaces using distributivity theory.

Findings

Categorical distributivity proves circuit equivalence and efficiency.

The approach generalizes quantum algorithm implementation to broader categories.

Laplaza's coherence theory underpins the categorical proof of quantum circuit efficiency.

Abstract

This paper studies one of the best known quantum algorithms - Shor's factorisation algorithm - via categorical distributivity. A key aim of the paper is to provide a minimal set of categorical requirements for key parts of the algorithm, in order to establish the most general setting in which the required operations may be performed efficiently. We demonstrate that Laplaza's theory of coherence for distributivity provides a purely categorical proof of the operational equivalence of two quantum circuits, with the notable property that one is exponentially more efficient than the other. This equivalence also exists in a wide range of categories. When applied to the category of finite dimensional Hilbert spaces, we recover the usual efficient implementation of the quantum oracles at the heart of both Shor's algorithm and quantum period-finding generally; however, it is also applicable in a much wider range of settings.