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arXiv · 2512.24798

Non-Abelian Geometric Phases in Triangular Structures And Universal SU(2) Control in Shape Space

J. Dai·A. Molochkov·A. J. Niemi·J. Westerholm
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Abstract

We construct holonomic quantum gates for qubits that are encoded in the near-degenerate vibrational $E$-doublet of a deformable three-body system. Using Kendall's shape theory, we derive the Wilczek--Zee connection governing adiabatic transport within the $E$-manifold. We show that its restricted holonomy group is $\mathrm{SU}(2)$, implying universal single-qubit control by closed loops in shape space. We provide explicit loops implementing a $\pi/2$ 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-$X$ (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($6s$)--Cs($6s$)--Cs($nd_{3/2}$) Rydberg trimer in optical tweezers and specify operating conditions that suppress leakage out of the $E$-manifold.

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J. Dai, A. Molochkov, A. J. Niemi, J. Westerholm. 2025-12-31. Non-Abelian Geometric Phases in Triangular Structures And Universal SU(2) Control in Shape Space. https://arxiv.org/abs/2512.24798

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