Qubit Geometry through Holomorphic Quantization

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

doi:10.2478/qic-2025-0022

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

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
[ 1001 ]\left[ {\matrix{ 1 \hfill & 0 \hfill \cr 0 \hfill & 1 \hfill \cr } } \right]	ψ(z)	Every point on ℂP¹	Every point on ℂP¹
[ 0110 ]\left[ {\matrix{ 0 \hfill & 1 \hfill \cr 1 \hfill & 0 \hfill \cr } } \right]	ψ(1z)\psi \left( {{1 \over z}} \right)	{±1}	{ (11),(1−1) }\left\{ {\left( \matrix{ 1 \cr 1 \cr} \right),\left( \matrix{ 1 \cr - 1 \cr} \right)} \right\}
[ 0−ii0 ]\left[ {\matrix{ 0 & { - i} \cr i & 0 \cr } } \right]	ψ(−1z)\psi \left( { - {1 \over z}} \right)	{±i}	{ (0i),(−i0) }\left\{ {\left( \matrix{ 0 \cr i \cr} \right),\left( \matrix{ - i \cr 0 \cr} \right)} \right\}
[ 100−1 ]\left[ {\matrix{ 1 & 0 \cr 0 & { - 1} \cr } } \right]	–ψ(z)	{∞, 0}	{ (10),(01) }\left\{ {\left( \matrix{ 1 \cr 0 \cr} \right),\left( \matrix{ 0 \cr 1 \cr} \right)} \right\}
12[ 111−1 ]{1 \over {\sqrt 2 }}\left[ {\matrix{ 1 & 1 \cr 1 & { - 1} \cr } } \right]	ψ(z+1z−1)\psi \left( {{{z + 1} \over {z - 1}}} \right)	{1±2}\{ 1 \pm \sqrt 2 \}	{ (1±21) }\left\{ {\left( \matrix{ 1 \pm \sqrt 2 \cr 1 \cr} \right)} \right\}
[ 100i ]\left[ {\matrix{ 1 \hfill & 0 \hfill \cr 0 \hfill & i \hfill \cr } } \right]	ψ(iz)	{0}	{ (10),(0i) }\left\{ {\left( \matrix{ 1 \cr 0 \cr} \right),\left( \matrix{ 0 \cr i \cr} \right)} \right\}
[ 100eiπ/4 ]\left[ {\matrix{ 1 & 0 \cr 0 & {{e^{i\pi /4}}} \cr } } \right]	12ψ(z+iz){1 \over {\sqrt 2 }}\psi (z + iz)	{0}	{ (10),(0eiπ/4) }\left\{ {\left( \matrix{ 1 \cr 0 \cr} \right),\left( \matrix{ 0 \cr {e^{i\pi /4}} \cr} \right)} \right\}
[ cos θ2−i sin θ2−i sin θ2cos θ2 ]\left[ {\matrix{ {\cos {\theta \over 2}} & { - i\sin {\theta \over 2}} \cr { - i\sin {\theta \over 2}} & {\cos {\theta \over 2}} \cr } } \right]	ψ(cos θ12z+i sin θ12i sin θ12z+cos θ12)\psi \left( {{{\cos {{{\theta _1}} \over 2}z + i\sin {{{\theta _1}} \over 2}} \over {i\sin {{{\theta _1}} \over 2}z + \cos {{{\theta _1}} \over 2}}}} \right)	{±1}	Similar as X-gate
[ cos θ2−sin θ2sin θ2cos θ2 ]\left[ {\matrix{ {\cos {\theta \over 2}} & { - \sin {\theta \over 2}} \cr {\sin {\theta \over 2}} & {\cos {\theta \over 2}} \cr } } \right]	ψ(cos θ22z−sin θ22sin θ22z+cos θ22)\psi \left( {{{\cos {{{\theta _2}} \over 2}z - \sin {{{\theta _2}} \over 2}} \over {\sin {{{\theta _2}} \over 2}z + \cos {{{\theta _2}} \over 2}}}} \right)	{±i}	Similar as Y-gate
[ e−iθ200eiθ2 ]\left[ {\matrix{ {{e^{{{ - i\theta } \over 2}}}} & 0 \cr 0 & {{e^{{{i\theta } \over 2}}}} \cr } } \right]	ψ(e^iθ₃z)	{∞, 0}	Similar as Z-gate

Qubit Geometry through Holomorphic Quantization

Figures & Tables

Figure 1.

Figure 2.

Figure 3.

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

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