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Non-Abelian geometric phases in triangular structures and universal SU(2) control in shape space

J. Dai1,*, A. Molochkov2,†, A. J. Niemi3,4,‡, and J. Westerholm5,§

  • *Contact author: jindai@bit.edu.cn
  • †Contact author: molochkov.alexander@gmail.com
  • ‡Contact author: Antti.Niemi@su.se
  • §Contact author: Jan.Westerholm@abo.fi
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Phys. Rev. A 114, 012212 – Published 20 July, 2026

DOI: https://doi.org/10.1103/wk9v-xp4l

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 that governs 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(nd3/2) Rydberg trimer in optical tweezers and specify operating conditions that suppress leakage out of the E manifold.

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Phys. Rev. A 114, 012212– Published 20 July, 2026
  • Received 22 October 2025
  • Accepted 11 June 2026
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