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Nature Communications

In-memory multilevel control of generic SO(m) holonomy in photonics

Youlve Chen, Jiaxin Zhang, Jinlong Xiang, An He, Weibiao Chen, Yikai Su, Junying Li, Xuhan Guo

Published Feb 7, 2026• 2 citations
Publisher

Abstract

Non-Abelian geometric phases offer fault-tolerant unitary operations that are essential for holonomic quantum computation. While the “all-geometric” approach has successfully enabled robust control of atoms, ions, and electrons in various quantum systems, its photonic implementations have remained limited by a lack of tunability—an essential feature for modern classical or quantum optical information processing. Here, we demonstrate nonvolatile multilevel tunable generic special-orthogonal (SO) geometric phases on a multilayer silicon photonic platform incorporating the phase-change material (PCM) of Sb₂Se₃. By toggling Sb₂Se₃ between crystalline and amorphous states, we dynamically control the number of degenerate states involved in the holonomy, enabling switching between different holonomic paths and SO(m) geometric transformations. Our results bridge the key gap between robust geometric-phase control and in-memory reconfigurable photonics, establishing a paradigm for fault-tolerant, high-dimensional operations in integrated optical systems. The researchers showcase nonvolatile multilevel tunable SO(m) holonomy in heterogeneous integrated photonics, merging static geometric-phase control with programmability for robust and adaptable “all-geometric” classical and quantum information processing.

DOI
10.1038/s41467-026-69287-2

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Non-Abelian geometric phases offer fault-tolerant unitary operations that are essential for holonomic quantum computation. While the “all-geometric” approach has successfully enabled robust control of atoms, ions, and electrons in various quantum systems, its photonic implementations have remained limited by a lack of tunability—an essential feature for modern classical or quantum optical information processing. Here, we demonstrate nonvolatile multilevel tunable generic special-orthogonal (SO) geometric phases on a multilayer silicon photonic platform incorporating the phase-change material (PCM) of Sb₂Se₃. By toggling Sb₂Se₃ between crystalline and amorphous states, we dynamically control the number of degenerate states involved in the holonomy, enabling switching between different holonomic paths and SO(m) geometric transformations. Our results bridge the key gap between robust geometric-phase control and in-memory reconfigurable photonics, establishing a paradigm for fault-tolerant, high-dimensional operations in integrated optical systems. The researchers showcase nonvolatile multilevel tunable SO(m) holonomy in heterogeneous integrated photonics, merging static geometric-phase control with programmability for robust and adaptable “all-geometric” classical and quantum information processing.
The non-Abelian geometric phase possesses the capability of enabling robust and fault-resilient unitary transformations, making it a cornerstone of holonomic quantum computation. This "all-geometric" approach has successfully advanced the manipulation of electrons in condensed matter physics and has sparked growing interest in its implementation within photonics, an area that has traditionally relied on sensitive dynamic phases. However, a major limitation of the topologically protected and inherently robust geometric phase is its lack of reconfigurability. In contrast, mainstream optical computing schemes demand high reconfigurability to compensate for fabrication errors and to support diverse computational tasks. Here, we demonstrate a reconfigurable non-Abelian geometric phase based on the non-volatile phase-change material Sb2_22​Se3_33​. By switching between its crystalline and amorphous states, the number of degenerate subspaces can be actively adjusted. Thus, multilevel second-order matrices and reconfigurable third-order matrices with 3-bit control is realized. For larger reconfigurable rotation angles, tunable braiding operations are also demonstrated. Furthermore, high-dimensional reconfigurable braiding shows promising potential for applications in optical switching. Our results pave the way for the all-geometric-phase-based approach in optical computing.
Non Abelian holonomy, a geometric aspect of quantum mechanics, underpins the geometric formulation of fundamental interactions and plays a vital role in holonomic quantum computation. Photonics is a fertile platform for exploring non-Abelian holonomies. However, universal unitary holonomies are still out of reach. The lack of control over scale phase degrees of freedom hinders the direct quantum simulation of unitary-governed phenomena, such as electron spin, weak isospin, and quantum chromodynamics,as well as the realization of generic geometric quantum gates for holonomic quantum computing. Here we employ artificial gauge fields (AGFs) to introduce complex-valued couplings, synthesizing arbitrary unitary holonomies and enabling generic geometric quantum gates in photonics. Higher dimensional U(m) transformations are experimentally synthesized based on Givens rotations. Our results pave the way for all-geometric-phase approach for both quantum and classical computing in photonics.
Abstract In this study we propose and numerically validate a protocol for high-fidelity non-Abelian holonomic gates in Si/SiGe spin-valley qubits, leveraging geometric phases for inherent fault tolerance. By combining electrical control of valley state energy splitting with cyclic in-plane magnetic fields, we engineer a protected subspace where quantum information evolves via geometric phases. Our simulations show these gates achieve 99.4% fidelity and maintain robustness against charge noise, outperforming conventional dynamic gates by 15% under realistic conditions. The gate exploits cyclic modulation of in-plane magnetic fields while dynamically tuning valley splitting to maintain Valley-Zeeman matching, yielding a Berry connection that enables non-commutative holonomies. Numerical analysis reveals distinct operational regimes, from hybridized states for phase accumulation to isolated states for initialization, within experimentally accessible parameter ranges. This work establishes spin-valley qubits as a promising platform for geometric quantum computation, combining holonomic operations' noise resilience with silicon's scalability. The demonstrated performance and compatibility with existing architectures suggest a viable path toward large-scale, fault-tolerant quantum processors.
We show that the notion of generalized Berry phase i.e., non-abelian holonomy, can be used for enabling quantum computation. The computational space is realized by a nnn-fold degenerate eigenspace of a family of Hamiltonians parametrized by a manifold calM\\cal McalM. The point of calM\\cal McalM represents classical configuration of control fields and, for multi-partite systems, couplings between subsystem. Adiabatic loops in the control manifold calM\\cal McalM induce non trivial unitary transformations on the computational space. For a generic system it is shown that this mechanism allows for universal quantum computation by composing a generic pair of loops in calM.\\cal M.calM.
Implementing quantum gates as non-Abelian holonomies, a class of topologically protected unitary operators, is a particularly promising paradigm for the design of intrinsically stable quantum computers [1]. In contrast to dynamic phases, the geometric phase accumulated by a quantum system propagating through a Hilbert spaceH\mathcal{H}Hdepends exclusively on its path. In general, geometric phases can exhibit arbitrary dimensionality. Wilczek and Zee introduced the idea of multi-dimensional, non-Abelian geometric phases - so called holonomies [2]. Anandan later dropped the requirement of adiabaticity to create holonomies, that are truly time-independent [3]. Non-adiabatic holonomies rely on a subspaceHgeo\mathcal{H}_{\text{geo}}Hgeo​of the Hilbert-space that is spanned by states{∣Φk⟩}k\{\vert \Phi_{k}\rangle\}_{k}{∣Φk​⟩}k​that fulfill(Φk∣H^∣Φj⟩=0(\Phi_{k}\vert \hat{H}\vert \Phi_{j}\rangle=0(Φk​∣H^∣Φj​⟩=0, whereH^\hat{H}H^is the system's Hamiltonian. Restricting the propagation toHgeo\mathcal{H}_{\text{geo}}Hgeo​ensures parallel transport and, thus, a purely geometric phase (see Fig. 1a) [4], [5]. Quantum optics constitutes a particularly versatile platform for quantum information processing, and in particular for the construction of non-adiabatic holonomic quantum computers: In addition to integration and miniaturization provided by the platform, the bosonic nature of photons also conveniently allows for multiple excitations of the same mode, readily expandingHgeo\mathcal{H}_{\text{geo}}Hgeo​and enabling the synthesis of holonomies from higher symmetry groupsU(N)\mathrm{U}(N)U(N)as larger and more capable computational units [6], [7].