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SU(3) / ℂP² Geometry for Qutrit Control and Holonomic Quantum Gates

Trent Slade

Published Dec 3, 2025• 0 citations
Publisher

Abstract

This report synthesizes theoretical and experimental work on SU(3) symmetry and ℂP² geometry as control manifolds for qutrit quantum computation. It consolidates explicit constructions of Hamiltonians, Berry/Wilczek–Zee connections, and holonomy operators across platforms including superconducting transmons, photonic orbital-mode qutrits, trapped ions, and ultracold atoms. Special emphasis is placed on geometric robustness, non-Abelian holonomies, and the triadic interference structures unique to SU(3). The review unifies Lie algebra geometry, fiber bundle theory, and quantum engineering into a coherent framework for high-dimensional holonomic control.

DOI
10.5281/zenodo.17798731

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This report synthesizes theoretical and experimental work on SU(3) symmetry and ℂP² geometry as control manifolds for qutrit quantum computation. It consolidates explicit constructions of Hamiltonians, Berry/Wilczek–Zee connections, and holonomy operators across platforms including superconducting transmons, photonic orbital-mode qutrits, trapped ions, and ultracold atoms. Special emphasis is placed on geometric robustness, non-Abelian holonomies, and the triadic interference structures unique to SU(3). The review unifies Lie algebra geometry, fiber bundle theory, and quantum engineering into a coherent framework for high-dimensional holonomic control.
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We present a geometric framework for holonomic quantum computing in which quantum gates arise from global properties of control manifolds rather than fine-tuned dynamical evolution. Quantum states are modeled as complex projective fibers over a classical control manifold, and adiabatic loops induce unitary gates through Berry and Wilczek–Zee holonomy. Within this setting, we introduce _Quantum Inner State Manifolds_ (QISMs) as symplectic fiber bundles equipped with a natural unitary connection governed by the Fubini–Study form. Using the Ambrose–Singer theorem, we show that generic QISMs generate holonomy groups dense in U(N)U{(N)}U(N), establishing universality. Fault tolerance emerges from global geometric features, providing a robust geometric foundation for quantum gate design.
We construct holonomic quantum gates for qubits that are encoded in the near-degenerate vibrational EEE-doublet of a deformable three-body system. Using Kendall's shape theory, we derive the Wilczek--Zee connection governing adiabatic transport within the EEE-manifold. We show that its restricted holonomy group is SU(2)\mathrm{SU}(2)SU(2), implying universal single-qubit control by closed loops in shape space. We provide explicit loops implementing a π/2π/2π/2 phase gate and a Hadamard-type gate. For two-qubit operations, we outline how linked holonomic cycles in arrays generate a controlled Chern--Simons phase, enabling an entangling controlled-XXX (CNOT) gate. We present a Ramsey/echo interferometric protocol that measures the Wilson loop trace of the Wilczek--Zee connection for a control cycle, providing a gauge-invariant signature of the non-Abelian holonomy. As a physically realizable demonstrator, we propose bond-length modulations of a Cs(6s6s6s)--Cs(6s6s6s)--Cs(nd3/2nd_{3/2}nd3/2​) Rydberg trimer in optical tweezers and specify operating conditions that suppress leakage out of the EEE-manifold.
A unifying framework for the control of quantum systems with non-Abelian holonomy is presented. It is shown that, from a control theoretic point of view, holonomic quantum computation can be treated as a control system evolving on a principal fiber bundle. An extension of methods developed for these classical systems may be applied to quantum holonomic systems to obtain insight into the control properties of such systems and to construct control algorithms for two established examples of the computing paradigm.
A unifying framework for the control of quantum systems with non-Abelian holonomy is presented. It is shown that, from a control theoretic point of view, holonomic quantum computation can be treated as a control system evolving on a principal fiber bundle. An extension of methods developed for these classical systems may be applied to quantum holonomic systems to obtain insight into the control properties of such systems and to construct control algorithms for two established examples of the computing paradigm.
This work develops Spectral Algebraics, a unified framework connecting the continuous spectrum of the Laplace–Beltrami operator on curved manifolds to the discrete algebra of SU(3) qutrit systems. By treating CP² and its associated line bundles as the geometric substrate of three-state quantum logic, the analysis shows that qutrit basis states emerge naturally as the lowest-energy sections of the connection Laplacian, while triadic interactions correspond to SU(3)’s symmetric d-tensor via geometric mode-overlap integrals.The manuscript constructs this bridge from spectral geometry, heat-kernel asymptotics, and representation theory through to field-theoretic interactions and spectral flow. The result is a programmable dictionary between geometric deformation and qutrit dynamics, suggesting that computation may be engineered through spectral shaping of cavities and metamaterials. The work provides a complete theoretical foundation for implementing qutrit hardware using Laplace–Beltrami eigenmodes.