Qubit geometry and conformal mapping

TL;DR

This paper explores the relationship between single qubit unitary operations and Möbius transformations by identifying the Bloch sphere with the Riemann sphere, revealing geometric and conformal mapping insights.

Contribution

It establishes a novel connection between quantum gate operations and conformal mappings on the extended complex plane, enriching the geometric understanding of qubit transformations.

Findings

Single qubit unitaries correspond to Möbius transformations.

The Bloch sphere is identified with the Riemann sphere.

New geometric relations between quantum operations and complex analysis.

Abstract

Identifying the Bolch sphere with the Riemann sphere(the extended complex plane), we obtain relations between single qubit unitary operations and M\"{o}bius transformations on the extended complex plane.