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

J. Dai, A. Molochkov, A. J. Niemi, J. Westerholm

TL;DR

The paper develops a universal holonomic quantum-control framework based on non-Abelian geometric phases in deformable three-body systems using Kendall's shape space. It derives the Wilczek–Zee connection over Kendall's shape sphere, proving the restricted holonomy is $SU(2)$ and enabling universal single-qubit gates via closed loops, with the gauge-invariant Wilson-loop trace as a diagnostic. It provides explicit single-qubit gates (a $\pi/2$ phase gate about $z$ and a Hadamard-type gate) and outlines a two-qubit CNOT gate via linked holonomic cycles and a Chern–Simons description of entangling phases, together with a concrete Cs trimer demonstrator and Ramsey–echo readout protocol. Finally, it discusses experimental considerations, potential for nonadiabatic shortcuts, and speculative connections to broader topological aspects such as the proton spin problem, highlighting shape-space holonomy as a robust route to geometric quantum control in few-body systems.

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 $π/2$ 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.

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

TL;DR

The paper develops a universal holonomic quantum-control framework based on non-Abelian geometric phases in deformable three-body systems using Kendall's shape space. It derives the Wilczek–Zee connection over Kendall's shape sphere, proving the restricted holonomy is and enabling universal single-qubit gates via closed loops, with the gauge-invariant Wilson-loop trace as a diagnostic. It provides explicit single-qubit gates (a phase gate about and a Hadamard-type gate) and outlines a two-qubit CNOT gate via linked holonomic cycles and a Chern–Simons description of entangling phases, together with a concrete Cs trimer demonstrator and Ramsey–echo readout protocol. Finally, it discusses experimental considerations, potential for nonadiabatic shortcuts, and speculative connections to broader topological aspects such as the proton spin problem, highlighting shape-space holonomy as a robust route to geometric quantum control in few-body systems.

Abstract

We construct holonomic quantum gates for qubits that are encoded in the near-degenerate vibrational -doublet of a deformable three-body system. Using Kendall's shape theory, we derive the Wilczek--Zee connection governing adiabatic transport within the -manifold. We show that its restricted holonomy group is , implying universal single-qubit control by closed loops in shape space. We provide explicit loops implementing a 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- (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()--Cs()--Cs() Rydberg trimer in optical tweezers and specify operating conditions that suppress leakage out of the -manifold.
Paper Structure (18 sections, 90 equations, 2 figures)

This paper contains 18 sections, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Wilczek-Zee connection
  3. Holonomic gates
  4. examples of gates
  5. Single qubit gates
  6. $\pi/2$ phase gate about the $z$ axis
  7. Hadamard gate from a rotation about the $y$ axis
  8. Two-qubit holonomic CNOT gate
  9. Outlook:
  10. Painléve's theorem
  11. Application to Cesium trimer
  12. Setup:
  13. Control knobs and mapping to $(A,\psi)$:
  14. A demonstrator loop and expected gate:
  15. Timing and adiabaticity:
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Panel a) A representative shape space time evolution of triangle $\triangle(t)$ with numerical parameter values $d=1.0 \, \& \, a=0.15 \, \& \, d_{12} = 1.1 \, \& \, a_{12} = 0.2 \, \& \, \Omega / \Omega_{12} = 3 \, \& \, m_{1,2} = 2.1 \, \& \, m_3=4.7$. Panel b) Corresponding rotational motion in the physical space. Panel c) Time evolutions of the angle $\theta(t)$ and the effective angular momentum (\ref{['eff']}). Panel d) Comparison of the rotational distance (effective angular velocity) with different values of $\phi_{13} - \phi_{23}$.
  • Figure 2: Rendering of proposed Cs Rydberg trimer in optical tweezers. Three optical tweezers, focused by a high-NA objective lense, trap three Cs atoms in a triangular configuration, realizing a Cs($6s$)--Cs($6s$)--Cs($nd_{3/2}$) Rydberg trimer. Rydberg excitation is illustrated schematically by a two-photon beam pair addressing the central site. (Not to scale.)