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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>E</a:mi> </a:math> doublet of a deformable three-body system. Using Kendall's shape theory, we derive the Wilczek-Zee connection that governs adiabatic transport within the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>E</b:mi> </b:math> manifold. We show that its restricted holonomy group is <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>SU</c:mi> <c:mo>(</c:mo> <c:mn>2</c:mn> <c:mo>)</c:mo> </c:mrow> </c:math> , implying universal single-qubit control by closed loops in shape space. We provide explicit loops implementing a <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>π</d:mi> <d:mo>/</d:mo> <d:mn>2</d:mn> </d:mrow> </d:math> 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- <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>X</e:mi> </e:math> () 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( <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mn>6</f:mn> <f:mi>s</f:mi> </f:mrow> </f:math> )-Cs( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mn>6</g:mn> <g:mi>s</g:mi> </g:mrow> </g:math> )-Cs( <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mi>n</h:mi> <h:msub> <h:mi>d</h:mi> <h:mrow> <h:mn>3</h:mn> <h:mo>/</h:mo> <h:mn>2</h:mn> </h:mrow> </h:msub> </h:mrow> </h:math> ) Rydberg trimer in optical tweezers and specify operating conditions that suppress leakage out of the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>E</i:mi> </i:math> manifold.