Toponomic Quantum Computation

TL;DR

This paper introduces toponomic quantum computation, utilizing topologically invariant non-abelian holonomies for robust quantum gates, demonstrated through explicit examples of topological NOT and CNOT gates.

Contribution

It identifies a class of subspaces with topologically invariant holonomies and provides explicit toponomic quantum gate constructions using a Majorana-like stellar representation.

Findings

Topological invariance of certain non-abelian holonomies under $SO(3)$ perturbations.

Explicit construction of topological NOT and CNOT gates.

Introduction of a Majorana-like stellar representation for subspaces.

Abstract

Holonomic quantum computation makes use of non-abelian geometric phases, associated to the evolution of a subspace of quantum states, to encode logical gates. We identify a special class of subspaces, for which a sequence of rotations results in a non-abelian holonomy of a topological nature, so that it is invariant under any $SO(3)$-perturbation. Making use of a Majorana-like stellar representation for subspaces, we give explicit examples of topological-holonomic (or toponomic) NOT and CNOT gates.