Holomorphic Quantization

Ingrid C. Daubechies

Ahmad Hazazi Ahmad Sumadi, Nurisya Mohd Shah, Umair Abdul Halim +1 more

Abstract We develop a wave mechanics formalism for qubit geometry using holomorphic functions and Möbius 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 Möbius action on holomorphic wavefunction. These wavefunctions emerge via a quantization process, with the Riemann sphere serving as the classical phase space of the qubit geometry. We quantize this space using Isham’s 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 on quantum information processing. This result provides a new direction for exploring quantum computation through Isham’s canonical group quantization and its holomorphic polarization method.

Gregory G. Athanasiu, Emmanuel G. Floratos, Stam Nicolis

We explicitly construct the quantization of classical linear maps of on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that finite quantum mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of to the subgroup , being the principal congruent subgroup , on a finite dimensional Hilbert space. The generators of the `rotation group' , , for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.

Gregory G. Athanasiu, Emmanuel G. Floratos, Stam Nicolis

We construct explicitly the quantization of classical linear maps of SL(2,{\\Bbb R}) on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of SL(2,{\\Bbb Z}) to the subgroup SL(2,{\\Bbb Z})/\\Gamma_l, \\Gamma_l being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, O_{l}(2)\\subset SL(2,l), for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.