A unified theory of quantum holonomies

H.C. Lee, Zhenduo Zhu

Vera Neef, Matthias Heinrich, Tom A. W. Wolterink +1 more

Holonomies are of great interest to quantum computation and simulation. The geometrical nature of holonomies offers increased stability to quantum gates. Furthermore, symmetries of particle physics are naturally reflected in holonomies, making them ideally suited for quantum simulation of quantum chromodynamics and grand unified theories. Yet, practically designing quantum holonomies with the required properties and scale is challenging. Here, we construct a new class of holonomies by increasing the particle number. We show that multiparticle holonomies can even exist in systems devoid of any single-particle holonomies. We present a comprehensive framework for multiparticle quantum holonomies and experimentally realize various two-particle holonomies in integrated photonics. Our results enable the number of particles to be harnessed as a design parameter, offering increased freedom in constructing holonomic systems.

Taksu Cheon, Atushi Tanaka, Sanguk Kim

James, Kazadi

Abstract: This work develops a unified and gauge-invariant framework in which quantum interference is described as a measurement of relative holonomy between two coherent evolutions in a combined configuration space X = M × Λ. Both the Aharonov–Bohm (Abelian) and Wilczek–Zee (non-Abelian) effects appear as limits of a single Wilson-loop observable. A block connection is introduced on the direct-sum bundle E = H ⊕ (T X ⊗ ℂ), with structure group U(n) × O(p,q), thereby unifying Berry parallel transport and Levi-Civita geometry within one connection. The formalism ensures gauge and diffeomorphism covariance, analytic well-posedness under back-reaction, and establishes conditions for reconstructing the spacetime metric from holonomy data. The resulting theory provides a mathematically consistent and experimentally testable synthesis of geometric phase, interference, and spacetime connection. Potential physical implementations include mesoscopic rings, cold-atom systems, and superconducting circuits. Keywords: geometric phase; holonomy; Aharonov–Bohm effect; Berry phase; Wilczek–Zee connection; differential geometry; quantum interference; gauge theory.

Siye Wu

Taksu Cheon, Atushi Tanaka

We present a unified formulation of quantum holonomy which is capable of describing all of its known varieties. The full role of non-Abelian gauge connection is elucidated. As examples, solvable models of quantum kicked spin are analyzed. They are shown to exhibit spiral holonomy, as well as Berry phase and Wilczek-Zee holonomy.