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

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 $\mathrm{S}\mathrm{U}(2)$, implying universal single-qubit control by closed loops in shape space. We provide explicit loops implementing a $\pi \mathrm{/}2$ 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($n{d}_{3\mathrm{/}2}$) Rydberg trimer in optical tweezers and specify operating conditions that suppress leakage out of the $E$-manifold.