Quantum information with general quantum variables: a formalism encompassing qubits, qudits, and quantum continuous variables

TL;DR

This paper introduces a unified formalism for quantum information that seamlessly integrates qubits, qudits, and continuous variables, enabling broad applicability of quantum computation results across different system types.

Contribution

It develops a simple, comprehensive framework that generalizes key quantum structures like Pauli operators, Clifford groups, and quantum gates for all dimensions.

Findings

Unified formalism for all quantum variable types

Generalized Pauli and Clifford groups for arbitrary dimensions

Universal gate sets applicable across systems

Abstract

This note presents a simple and unified formulation of the most fundamental structures used in quantum information with qubits, arbitrary dimension qudits, and quantum continuous variables. This \emph{general quantum variables} construction provides a succinct language for formulating many results in quantum computation and information so that they are applicable in all dimensions. The structures included within this formalism include: a generalization to arbitrary dimension of the three Pauli operators, and the associated mutually unbiased bases; the Pauli and Clifford groups; many important quantum gates; standard sets of generators for the Clifford group; and simple universal gate sets. This formalism provides a convenient, intuitive and extensible language for easily generalizing results that were originally derived for a single type of system (often qubits or quantum continuous variables), and that rely on only those structures listed above, to apply in all dimensions.