Quantum Information & Computation
Volume 25, Issue 5
RESEARCH ARTICLE Open access

Qubit Geometry through Holomorphic Quantization

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

Author affiliations
  • Ahmad Hazazi Ahmad Sumadi1
  • Nurisya Mohd Shah2
  • Umair Abdul Halim3
  • Hishamuddin Zainuddin4
  1. Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia (UPM), UPM Serdang, Selangor Darul Ehsan, Selangor Malaysia
  2. Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia (UPM), UPM Serdang, Selangor Darul Ehsan, Selangor Malaysia; Department of Physics, Universiti Putra Malaysia (UPM), UPM Serdang, Selangor Darul Ehsan, Selangor Malaysia
  3. Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia (UPM), UPM Serdang, Selangor Darul Ehsan, Selangor Malaysia; Center for Foundation Studies in Science of Universiti Putra Malaysia (ASPutra), Universiti Putra Malaysia (UPM), UPM Serdang, Selangor Darul Ehsan, Selangor Malaysia
  4. School of Mathematics and Physics, Xiamen University Malaysia, Jalan Sunsuria, Bandar Sunsuria, Sepang, Selangor Darul Ehsan, Malaysia

pp. 385-396 · 10.2478/qic-2025-0022

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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.
Open accessCC BY-NC-ND 4.0

Keywords

  • canonical group quantization
  • compact phase space
  • holomorphic polarization
  • qubit geometry
  • qubit operations

Suggested citation

Ahmad Hazazi Ahmad Sumadi, Nurisya Mohd Shah, Umair Abdul Halim, Hishamuddin Zainuddin (2025). Qubit Geometry through Holomorphic Quantization. Quantum Information & Computation, 25(5), 385-396. https://doi.org/10.2478/qic-2025-0022

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References (47)

  1. P. W. Shor (1994). Proceedings 35th Annual Symposium on Foundations of Computer Science (Santa Fe, NM, USA), 124–134.Google Scholar
  2. A. W. Harrow and A. Montanaro (1997). “Quantum computational supremacy.” Nature, 549, 14.DOI: 10.1038/nature23458Google Scholar
  3. S. Wiesner (1983). “Conjugate coding.” ACM Sigact News, 15, 1.DOI: 10.1145/1008908.1008920Google Scholar
  4. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters (1993). “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels.” Physical Review Letters, 70, 1895.DOI: 10.1103/PhysRevLett.70.1895Google Scholar
  5. V. Giovannetti, S. Llyod, and L. Maccone (2004). “Quantum-enhanced measurements: Beating the standard quantum limit.” Science, 306, 5700.DOI: 10.1126/science.1104149Google Scholar
  6. M. A Nielsen, and I.L. Chuang (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press.Google Scholar
  7. T. W. B. Kibble (1979). “Geometrization of quantum mechanics.” Communications in Mathematical Physics, 65, 189–201.DOI: 10.1007/BF01225149Google Scholar
  8. D. C. Brody, and L. P. Hughston, (2002). “Geometric quantum mechanics.” Journal of Geophysics, 38, 1.DOI: 10.1016/S0393-0440(00)00052-8Google Scholar
  9. J. Anandan, and Y. Aharonov, (1990). “Geometry of quantum evolution.” Physical Review Letters, 65, 1697.DOI: 10.1103/PhysRevLett.65.1697Google Scholar
  10. A. Ashtekar, and T. A Schilling, (1995). “Geometry of quantum mechanics.” AIP Conference Proceedings, 342, 1.DOI: 10.1063/1.48786Google Scholar
  11. E. Ercolessi, R. Fioresi, and T. Weber (2024). “The geometry of quantum computing.” International Journal of Geometric Methods in Modern Physics, 21, 10.DOI: 10.1142/S0219887824400115Google Scholar
  12. R. Mosseri, and R. Dandoloff (2001). “Geometry of entangled states, bloch sphere and hopf fibrations.” Journal of Physics A: Mathematical and General, 34, 10243.DOI: 10.1088/0305-4470/34/47/324Google Scholar
  13. R. Mosseri (2006). Topology in Condensed Matter Physic. Springer Berlin Heidelberg New York, 87–203.Google Scholar
  14. I.Bengtsson and K. Zyczkowski (2017). Geometry of Quantum States - An Introduction to Quantum Entanglement 2nd ed. Cambridge University Press.DOI: 10.1017/9781139207010Google Scholar
  15. I. Bengtsson, J. Brännlund, and K. Zyczkowsi (2002). “CPn or entanglement illustrated.” International Journal of Modern Physics A, 17, 31.DOI: 10.1142/S0217751X02010820Google Scholar
  16. F. Frescura and B. Hiley (1981). “Geometric interpretation of The Pauli Spinor.” American Journal of Physics, 49, 152.DOI: 10.1119/1.12548Google Scholar
  17. M. Kus, and K. Zyczkowski (2001). “Geometry of entangled states.” Physical Review A, 63, 032307.DOI: 10.1103/PhysRevA.63.032307Google Scholar
  18. P. Levay (2004). “The geometry of entanglement: metrics, connections and the geometric phase.” Journal of Physics A: Mathematical and General, 37, 1821.DOI: 10.1088/0305-4470/37/5/024Google Scholar
  19. J. M. Isidro (2004). “Quantum states from tangent vectors.” Modern Physics Letters A, 19, 31.DOI: 10.1142/S0217732304015634Google Scholar
  20. R. Loll (1992). Mathematical Aspects of Classical Field Theory (AMS), pp. 503–530.DOI: 10.1090/conm/132/1188455Google Scholar
  21. E. Witten (1991). “Geometric quantization of Chern-Simons Gauge theory.” Journal of Differential Geometry, 3, 3.Google Scholar
  22. U. Chabaud and S.Mehraban (2022). “Holomorphic representation of quantum computations,” Quantum, 6, 831.DOI: 10.22331/q-2022-10-06-831Google Scholar
  23. I. Bjelakovic and W. Stulpe (2005). “The projective Hilbert space as a classic phase space for nonrelativistic quantum dynamics.” International Journal of Theoretical Physics, 44.DOI: 10.1007/s10773-005-8982-2Google Scholar
  24. H. Urbantke (1991). “Two-level quantum systems: states, phases, and holonomy.” American Journal of Physics, 59, 6.DOI: 10.1119/1.16809Google Scholar
  25. J. W. Lee, et al. (2002). “Qubit geometry and conformal mapping.” Quantum Information Processing, 1.DOI: 10.1023/A:1019645000745Google Scholar
  26. C.J. Isham (1984). Relativity, Groups and Topology II, B.S. DeWitt and R. Stora (eds.). North-Holland, 1061–1290.Google Scholar
  27. N.M.J. Woodhouse (1997). Geometric Quantization, 2nd ed., Clarendon Press.Google Scholar
  28. C. J. Isham and A. C. Kakas (1984). “A group theoretical approach to the canonical quantisation of gravity. I. Construction of the canonical group.” Classical and Quantum Gravity, 1, 6.DOI: 10.1088/0264-9381/1/6/008Google Scholar
  29. C. J. Isham and A. C. Kakas (1984). “A group theoretical approach to the canonical quantisation of gravity. II. Unitary representations of the canonical group.” Classical and Quantum Gravity, 1, 6.DOI: 10.1088/0264-9381/1/6/009Google Scholar
  30. C. J. Isham and N. Linden (1988). “A group theoretic quantisation of strings on Tori.” Classical and Quantum Gravity, 5, 6.DOI: 10.1088/0264-9381/5/1/014Google Scholar
  31. H. Zainuddin (1989). “Group-theoretic Quantization of a Particle On a Torus in a Constant Magnetic Field.” Physical Review D, 40, 636.DOI: 10.1103/PhysRevD.40.636Google Scholar
  32. A. Bouketir (2000). “Group-theoretic quantisation on spheres and quantum hall effect.” (Unpublished) PhD thesis, Universiti Putra Malaysia, Malaysia.Google Scholar
  33. M. F. Umar, N. M. Shah and H. Zainuddin (2018). “Two-dimensional plane, modified symplectic structure and quantization.” Jurnal Fizik Malaysia, 39, 2.Google Scholar
  34. C. Benevides, and A. Reyes (2004). Geometric and Topological Methods for Quantum Field Theory. Cambridge University Press, 344–367.Google Scholar
  35. R. A. Silva and T. Jacobson (2021). “Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole.” Journal of Physics A: Mathematical and Theoretical, 54, 235303.DOI: 10.1088/1751-8121/abf961Google Scholar
  36. R. A. Silva and T. Jacobson (2023). “Causal diamonds in (2+1)-dimensional quantum gravity.” Physical Review D, 107, 2.Google Scholar
  37. C. Brif (1997). “SU (2) and SU (1,1) algebra eigenstates: A unified analytic approach to coherent and intelligent states.” International Journal of Theoretical Physics, 36.DOI: 10.1007/BF02435763Google Scholar
  38. K. Fujii (2002). arXiv:quant-ph/0112090.Google Scholar
  39. S. T. Ali and M. Englis (2005). “Quantization methods: A guide for physicists and analysts.” Reviews in Mathematical Physics, 17, 4.DOI: 10.1142/S0129055X05002376Google Scholar
  40. P. Andrews (1988). “The classification of surfaces.” The American Mathematical Monthly, 95, 9.DOI: 10.1080/00029890.1988.11972103Google Scholar
  41. I. Vaisman (1981). “The Bott obstruction to the existence of nice polarizations,” Minnesota Math, 92.DOI: 10.1007/BF01442487Google Scholar
  42. G.A. Jones and I. Singerman (1987). Complex Functions - An Algebraic and Geometric Viewpoint. Cambridge University Press.DOI: 10.1017/CBO9781139171915Google Scholar
  43. A.H.A. Sumadi (2024). “Canonical group quantization on non-cotangent bundle phase space and its application in quantum information theory.” (Unpublished) PhD thesis, Universiti Putra Malaysia.Google Scholar
  44. D. Gurarie (2008). Symmetries and Laplacians - Introduction to Harmonic Analysis, Group Representations and Applications. Dover.Google Scholar
  45. C.P. Boyer, E.G. Kalnins and P. Winternitz (1985). “Separation of variables for the Hamilton-Jacobi equation on complex projective spaces.” SIAM Journal on Mathematical Analysis, 16.DOI: 10.1137/0516006Google Scholar
  46. J. Mostovoy (1998). “Geometry of truncated symmetric products and real roots of real polynomials.” Bulletin of the London Mathematical Society, 30.DOI: 10.1112/S0024609397003883Google Scholar
  47. P. Blagojevic, V. Grujic and R. Zivaljevic (2004). arXiv:math/0408417v1.Google Scholar