Qubits from Adinkra Graph Theory via Colored Toric Geometry

TL;DR

This paper introduces a novel geometric framework linking toric geometry, Adinkra graphs, and qubit systems, enabling the use of geometric data to analyze and manipulate quantum information and gates.

Contribution

It establishes a one-to-one correspondence between qubit systems and specific toric varieties using colored toric geometry, providing a new geometric approach to quantum information theory.

Findings

Qubit systems correspond to specific toric varieties: CP^1, CP^1×CP^1, and CP^1×CP^1×CP^1.

Operations on toric data can generate universal quantum gates.

The approach offers a geometric perspective for understanding quantum information processing.

Abstract

We develop a new approach to deal with qubit information systems using toric geometry and its relation to Adinkra graph theory. More precisely, we link three different subjects namely toric geometry, Adinkras and quantum information theory. This one to one correspondence may be explored to attack qubit system problems using geometry considered as a powerful tool to understand modern physics including string theory. Concretely, we examine in some details the cases of one, two, and three qubits, and we find that they are associated with \bf CP^1, \bf CP^1\times CP^1 and \bf CP^1\times CP^1\times CP^1 toric varieties respectively. Using a geometric procedure referred to as colored toric geometry, we show that the qubit physics can be converted into a scenario handling toric data of such manifolds by help of Adinkra graph theory. Operations on toric information can produce universal quantum gates.