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

TL;DR

This paper demonstrates how non-Abelian geometric phases in a deformable three-body system enable universal quantum control, including single-qubit gates and entangling two-qubit operations, with a proposed physical implementation using Rydberg trimers.

Contribution

It introduces a method to realize universal SU(2) control via non-Abelian holonomies in shape space, with explicit gate constructions and a measurement protocol.

Findings

Holonomic gates for qubits encoded in vibrational modes

Explicit implementation of a $C0/2$ phase and Hadamard gates

Proposal for a Rydberg trimer system as a physical realization

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.