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 Sb$_2$Se$_3$. 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.
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 Sb2Se3. 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 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.
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.
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 spaceHdepends 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 subspaceHgeoof the Hilbert-space that is spanned by states{∣Φk⟩}kthat fulfill(Φk∣H^∣Φj⟩=0, whereH^is the system's Hamiltonian. Restricting the propagation toHgeoensures 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 expandingHgeoand enabling the synthesis of holonomies from higher symmetry groupsU(N)as larger and more capable computational units [6], [7].
Quantum computation has been shown to be superior to classical computation in solving some problems, and therefore can substantially change our life. However, the realization of quantum computation is still challenging, even if quantum technologies have been improved significantly. Two main obstacles to the realization of practical quantum computation are control errors and decoherence. Control errors are caused by inaccurate manipulations of quantum systems and decoherence is caused by the inevitable interaction between the system and its environment. Geometric phases are only dependent on evolution paths of quantum systems but independent of the evolution details and therefore quantum computation based on geometric phases, i.e., geometric quantum computation, is robust against control errors, benefiting the realization of practical quantum computation. The early proposals of geometric quantum computation are based on adiabatic geometric phases. These proposals require quantum systems to undergo adiabatic evolution, which makes quantum systems evolve for a long time. To circumvent this, nonadiabatic geometric quantum computation based on nonadiabatic Abelian geometric phases was proposed soon after. In 2012, nonadiabatic holonomic quantum computation based on nonadiabatic non-Abelian geometric phases was proposed, which also circumvents long-time evolutions. Moreover, compared with nonadiabatic geometric quantum computation that uses the geometric phase as one parameter of a quantum gate, nonadiabatic holonomic quantum computation uses the holonomic matrix itself as a quantum gate. This makes nonadiabatic holonomic quantum computation possess whole-geometric property. Due to the merits of both geometric robustness and high-speed implementation without the limit of adiabatic evolution, nonadiabatic holonomic quantum computation has been attracting much attention. Until now, much progress has been achieved in the field of nonadiabatic holonomic quantum computation. On one hand, various methods have been proposed to design more efficient nonadiabatic holonomic gates. Nonadiabatic holonomic gates were first realized by using resonant laser fields to drive a three-level system. After this, the single-shot proposal and the single-loop proposal were proposed, allowing us to realize one-qubit gates by a shorter path and thereby reducing the exposure time of nonadiabatic holonomic gates to the environment. To further shorten the exposure time, the path-shortening protocol was put forward, where nonadiabatic holonomic gates can be realized based on a class of extended evolution paths that are shorter than the former ones. Recently, a general approach of constructing Hamiltonians for nonadiabatic holonomic quantum computation was put forward, by using which one can easily find a Hamiltonian making the quantum system evolve along a desired path so that nonadiabatic holonomic gates can be realized with an economical evolution time. On the other hand, various methods have been proposed to combine nonadiabatic holonomic gates and various decoherence-resilient methods, making the resulting schemes robust against both control errors and decoherence. The first proposal in this aspect is combining nonadiabatic holonomic gates and decoherence-free subspaces. Gradually, proposals combining nonadiabatic holonomic gates with noiseless subsystems, dynamical decoupling and surface codes were put forward. Last but not least, various proposals suitable for specific physical systems have been proposed and particularly various experimental platforms have been used to demonstrate nonadiabatic holonomic gates. This also significantly improves the development of nonadiabatic holonomic quantum computation. In this paper, we review the above research advances on nonadiabatic holonomic quantum computation, aiming to help readers understand the main developments of nonadiabatic holonomic quantum computation.