The Coxeter symmetries of high-level general sequential computation in the invertible-digital and quantum domains

TL;DR

This paper explores how high-level invertible sequential processes in digital and quantum computing are governed by Coxeter groups, revealing their structure and representations in quantum systems.

Contribution

It demonstrates that both invertible digital and quantum sequential processes are characterized by Coxeter groups, providing explicit group presentations and unitary representations in quantum spaces.

Findings

Coxeter groups determine all processes from elemental sequences.

Quantum processes form unitary representations of Coxeter groups.

Explicit group presentations are provided for all cases.

Abstract

The article investigates high-level general invertible-sequential processing in the digital and quantum domains. In particular it is shown that (i) invertible digital-sequential processes, constructed using a standard general-inversion procedure, and (ii) sequential quantum processes, determine Coxeter groups. In each case the groups are seen to define all processes that may be constructed from the, given, elemental processes of the sequences. Explicit forms of the presentations of the Coxeter groups are given for all cases. The quantum processes are seen to define unitary representations of the associated Coxeter groups in tensor-product qubit spaces.