Qubit Geometry through Holomorphic Quantization

Ahmad Hazazi Ahmad Sumadi; Nurisya Mohd Shah; Umair Abdul Halim; Hishamuddin Zainuddin

doi:10.2478/qic-2025-0022

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Qubit Geometry through Holomorphic Quantization

Quantum Information & Computation

Volume 25 (2025): Issue 5 (December 2025)

By: Ahmad Hazazi Ahmad Sumadi, Nurisya Mohd Shah, Umair Abdul Halim and Hishamuddin Zainuddin

Open Access

|Dec 2025

Figures & Tables

Stereographic projection from the north pole onto the plane x₃ = 0.

Holomorphic wavefunction from local section of ℒ.

Fixed points of quantum gate wavefunction on ℂP¹

Table 1.

Geometric interpretation of quantum gates via their holomorphic wavefunctions, including their action on the Riemann sphere, fixed points, and induced transformations.

Common Gate	Gates’ Wavefunctions	Fixed Points	Corresponding Eigenstates
$[\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}]$	ψ(z)	Every point on ℂP¹	Every point on ℂP¹
$[\begin{array}{l} 0 & 1 \\ 1 & 0 \end{array}]$	$ψ (\frac{1}{z})$	{±1}	${(\begin{matrix} 1 \\ 1 \end{matrix}), (\begin{matrix} 1 \\ - 1 \end{matrix})}$
$[\begin{matrix} 0 & - i \\ i & 0 \end{matrix}]$	$ψ (- \frac{1}{z})$	{±i}	${(\begin{matrix} 0 \\ i \end{matrix}), (\begin{matrix} - i \\ 0 \end{matrix})}$
$[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$	–ψ(z)	{∞, 0}	${(\begin{matrix} 1 \\ 0 \end{matrix}), (\begin{matrix} 0 \\ 1 \end{matrix})}$
$\frac{1}{\sqrt{2}} [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}]$	$ψ (\frac{z + 1}{z - 1})$	${1 \pm \sqrt{2}}$	${(\begin{matrix} 1 \pm \sqrt{2} \\ 1 \end{matrix})}$
$[\begin{array}{l} 1 & 0 \\ 0 & i \end{array}]$	ψ(iz)	{0}	${(\begin{matrix} 1 \\ 0 \end{matrix}), (\begin{matrix} 0 \\ i \end{matrix})}$
$[\begin{matrix} 1 & 0 \\ 0 & e^{i π / 4} \end{matrix}]$	$\frac{1}{\sqrt{2}} ψ (z + i z)$	{0}	${(\begin{matrix} 1 \\ 0 \end{matrix}), (\begin{matrix} 0 \\ e^{i π / 4} \end{matrix})}$
$[\begin{matrix} \cos \frac{θ}{2} & - i \sin \frac{θ}{2} \\ - i \sin \frac{θ}{2} & \cos \frac{θ}{2} \end{matrix}]$	$ψ (\frac{\cos \frac{θ_{1}}{2} z + i \sin \frac{θ_{1}}{2}}{i \sin \frac{θ_{1}}{2} z + \cos \frac{θ_{1}}{2}})$	{±1}	Similar as X-gate
$[\begin{matrix} \cos \frac{θ}{2} & - \sin \frac{θ}{2} \\ \sin \frac{θ}{2} & \cos \frac{θ}{2} \end{matrix}]$	$ψ (\frac{\cos \frac{θ_{2}}{2} z - \sin \frac{θ_{2}}{2}}{\sin \frac{θ_{2}}{2} z + \cos \frac{θ_{2}}{2}})$	{±i}	Similar as Y-gate
$[\begin{matrix} e^{\frac{- i θ}{2}} & 0 \\ 0 & e^{\frac{i θ}{2}} \end{matrix}]$	ψ(e^iθ₃z)	{∞, 0}	Similar as Z-gate

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DOI: https://doi.org/10.2478/qic-2025-0022 | Journal eISSN: 3106-0544 (formerly 1533-7146) | Journal ISSN: 1533-7146

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Language: English

Page range: 385 - 396

Submitted on: May 13, 2025

Accepted on: Aug 3, 2025

Published on: Dec 31, 2025

Published by: Cerebration Science Publishing Co., Limited

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

canonical group quantization,

compact phase space,

holomorphic polarization,

qubit geometry,

qubit operations

Related subjects:

Computer sciences,

Computer sciences in the humanities,

Computer sciences in natural sciences,

Mathematics,

Applied mathematics,

Physics,

Quantum physics

© 2025 Ahmad Hazazi Ahmad Sumadi, Nurisya Mohd Shah, Umair Abdul Halim, Hishamuddin Zainuddin, published by Cerebration Science Publishing Co., Limited
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 25 (2025): Issue 5 (December 2025)

Qubit Geometry through Holomorphic Quantization

Figures & Tables

Figure 1.

Figure 2.

Figure 3.

Table 1.

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