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.
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.
In this work, we investigate the geometry of quantum logic gates within the holomorphic representation of quantum mechanics. We begin by embedding the physical qubit subspace into the space of holomorphic functions that are homogeneous of degree one in each Schwinger boson pair (zaj,zbj). Within this framework, we derive explicit closed-form differential operator representations for a universal set of quantum gates, including the Pauli operators, Hadamard, CNOT, CZ, and SWAP, and demonstrate that they preserve the physical subspace exactly. Restricting to unit-magnitude variables (∣z∣=1) reveals a toroidal space T2N, on which quantum gates act as canonical transformations: Pauli operators generate Hamiltonian flows, the Hadamard gate induces a nonlinear automorphism, and entangling gates produce correlated diffeomorphisms that couple distinct toroidal factors. Beyond the torus, the full Segal--Bargmann space carries a natural Kähler geometry that governs amplitude dynamics. Entanglement is geometrically characterized via the Segre embedding into complex projective space, while topological protection arises from the U(1)N fiber bundle structure associated with the Jordan--Schwinger constraint.
We extend the framework of the Three-Dimensional Complex Necessity (3DCN) principle to quantum information theory, demonstrating that the complex probability amplitudes of qubits are emergent manifestations of real, restricted geometric rotations on a 2D transverse phase space substrate. Conventionally, a quantum bit (qubit) is axiomatically formulated on a complex Hilbert space, leaving its state vulnerable to environmental decoherence. By replacing the algebraic primitive i with the coordinate-covariant universal complex structure tensor J_U satisfying J_U^2 = -I derived from oriented 3D geometry, we construct a purely real-geometric formulation of quantum computing. We mathematically prove that the canonical Bloch sphere can be mapped onto a real 4-dimensional physical phase space reduced by first-class constraints. Under this formulation, unitary quantum gates (such as Hadamard and CNOT) are executed via real spatial transformations, while the topological Ward identity governing the interaction locking mechanism provides a built-in geometric defense against phase decoherence without requiring active quantum error correction protocols.
This work reformulates quantum mechanics on the real plane (R2), establishing an algebraic isomorphism to the complex field (C) and demonstrating that the complex structure inherent to the Schrödinger equation can be fully captured by coupled real-valued dynamics. By constructing a field (R2,+,⋅) isomorphic to C, the real counterpart of the Schrödinger equation is derived, yielding a system of coupled real partial differential equations. This system reveals a symplectic structure where the symplectic matrix J geometrically represents the imaginary unit i, acting as a phase-space rotation operator. Time-independent solutions decouple into identical real Helmholtz equations, preserving spectral equivalence. The reformulation provides geometric insights into foundational aspects: the geometric representation of i, the emergence of probability conservation from real-component interactions, and the encoding of quantum phase relationships through the multiplicative structure of the R2 field. This perspective reinforces connections to classical Hamiltonian mechanics and geometric quantization, suggesting that complex numbers are structural, emerging from field operations on the real plane, rather than fundamental. Furthermore, quantum computation elements like qubits and gates are shown to have equivalent representations as real symplectic and special orthogonal matrices, maintaining unitary equivalence. We introduce a framework for quantum computation within a real vector space and provide a generalized protocol for generating entangled Bell states. These results offer a new perspective on the necessity of complex numbers in quantum theory, providing a framework with potential computational advantages for hybrid quantum-classical systems and enhanced geometric interpretability.
This paper explores the integration of quantum states into geometric transformation descriptions, aiming to revolutionize computation through the representation of geometric transformations. Traditional methods often rely on classical descriptions, but this work proposes a novel framework where quantum states directly encode geometric information, offering the potential for significantly more efficient and expressive computation. We detail the core mechanism, outlining the development of a system that maps geometric transformations to quantum states, enabling computational operations based on these quantum states. The paper introduces a conceptual model and preliminary results demonstrating the feasibility of this approach, highlighting its potential impact on various fields including geometry, computer graphics, and optimization. The core claim is that this framework facilitates efficient computation through the direct manipulation of quantum states, providing a fundamental shift in how geometric transformations are represented and processed.
Quantum mechanics is traditionally formulated in terms of complex Hilbert spaces, wavefunctions, and linear evolution equations. While extraordinarily successful, this framework leaves open several conceptual questions concerning the geometric origin of the wavefunction, the meaning of quantum phase, and the status of Hilbert space as a fundamental structure. In this work, we investigate a coherence-geometric reformulation in which the fundamental object is a coherence coordinate , where represents radial coherence and represents phase coherence associated with geometric transport. We show that the standard structures of quantum mechanics admit a natural interpretation within a differential-geometric framework. The wavefunction emerges from coherence coordinates, probability density is generated by radial coherence, probability flow is generated by phase coherence, and Hilbert spaces appear as spaces of square-integrable functions defined on a coherence manifold. The resulting formulation provides a unified geometric language connecting quantum mechanics, geometric phases, fiber bundles, holonomies, gauge theory, and differential geometry
Using Isham's group-theoretic quantization scheme, we construct the canonical groups of the systems on the two-dimensional sphere and one-dimensional complex projective space, which are homeomorphic. In the first case, we take SO(3) as the natural canonical Lie group of rotations of the two-sphere and find all the possible Hamiltonian vector fields, and followed by verifying the commutator and Poisson bracket algebra correspondences with the Lie algebra of the group. In the second case, the same technique is resumed to define the Lie group, in this case SU (2), of CP'.We show that one can simply use a coordinate transformation from S 2 to CP 1 to obtain all the Hamiltonian vector fields of CP 1 . We explicitly show that the Lie algebra structures of both canonical groups are locally homomorphic. On the other hand, globally their corresponding canonical groups are acting on different geometries, the latter of which is almost complex. Thus the canonical group for CP 1 is the double-covering group of SO(3), namely SU(2). The relevance of the proposed formalism is to understand the idea of CP 1 as a space of where the qubit lives which is known as a Bloch sphere.