Itinerant Quantum Integers: The Language of Quantum Computers

TL;DR

This paper introduces a novel framework using quantum integers to model qubits as pairs of compatible null vectors, providing new identities and representations for quantum computing calculations.

Contribution

It develops a new approach to quantum computation using real and complex quantum integers, linking null vectors to qubit states and offering algebraic identities and group representations.

Findings

Qubits modeled as pairs of compatible null vectors

Development of basic identities for quantum calculations

Two representations of the symmetric group provided

Abstract

The concept of positively and negatively compatible null vectors arises in the study of Clifford geometric algebras with a Lorentz-Minkowski metric. In previous works, the basic properties of such algebras have been set down in terms of a new principle of quantum duality. In the present work, the same structure is studied in terms of real and complex quantum integers, which generalize the real and complex number systems. It seems natural to identify a qubit as a pair of compatible null vectors; the up state of the qubit being their sum, and the down state being their difference. Basic identities are developed to make calculations routine, and two different representations of the symmetric group are given.