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 SU(2), implying universal single-qubit control by closed loops in shape space. We provide explicit loops implementing a π/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.