Geometric Phase-Driven Non-Abelian Holonomic Gates in Multivalley Silicon Spin Qubits for Fault-Tolerant Quantum Computing

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.