Qubit Geometry through Holomorphic Quantization

Abstract

We develop a wave mechanics formalism for qubit geometry using holomorphic functions and Mobius transformations, providing a geometric perspective on quantum computation. This framework extends the standard Hilbert space description, offering a natural interpretation of standard quantum gates on the Riemann sphere that is examined through their Mobius action on holomorphic wavefunction. These wavefunctions emerge via a quantization process, with the Riemann sphere serving as the classical phase space of qubit geometry. We quantize this space using canonical group quantization with holomorphic polarization, yielding holomorphic wavefunctions and spin angular momentum operators that recover the standard $SU(2)$ algebra with interesting geometric properties. Such properties reveal how geometric transformations induce quantum logic gates on the Riemann sphere, providing a novel perspective in quantum information processing. This result provides a new direction for exploring quantum computation through Isham's canonical group quantization and its holomorphic polarization method.

18 pages, 2 figures

References in corpus (13)

• Quantum-enhanced measurements: beating the standard quantum limit
• Quantum Computational Supremacy
• Geometric Quantum Mechanics
• Quantization Methods: A Guide for Physicists and Analysts
• Geometry of entangled states
• Geometry of entangled states, Bloch spheres and Hopf fibrations
• The geometry of entanglement: metrics, connections and the geometric phase
• SU(2) and SU(1,1) algebra eigenstates: A unified analytic approach to coherent and intelligent states
• CP^n, or, entanglement illustrated
• Holomorphic representation of quantum computations
• Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole
• Quantum States from Tangent Vectors
• The Geometry of Quantum Computing