An Algebraic Framework for Multi-Qudit Computations with Generalized Clifford Algebras

TL;DR

This paper introduces an algebraic framework for multi-qudit representations of generalized Clifford algebras, providing minimal assumptions and an explicit model, paving the way for algebraic graphical calculus development.

Contribution

It develops a minimal axiomatic algebraic framework and constructs an explicit model for multi-qudit generalized Clifford algebras, facilitating further algebraic and graphical methods.

Findings

Framework with minimal assumptions established

Explicit model satisfying the axioms constructed

Foundation for algebraic graphical calculus laid out

Abstract

In this article, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Strengths of our algebraic framework include the minimality of its assumptions, and the readiness by which one may give an explicit construction satisfying these assumptions. In terms of applications, this algebraic framework provides a solid foundation which opens the way for developing a graphical calculus for multi-qudit representations of generalized Clifford algebras using purely algebraic methods, which is addressed in a follow-up paper.