  Non-Abelian geometry phase has attracted significant attention for the robust
holonomic unitary behavior exhibited, which arises from the degenerate subspace
evolving along a trajectory in Hilbert space. It has been regarded as a
promising approach for implementing topologically protected quantum computation
and logic manipulation. However, due to the challenges associated with
high-dimensional parameters manipulation, this matrix-valued geometry phase has
not been realized on silicon integrated photonic platform, which is CMOS
compatible and regarded as the most promising flatform for next-generation
functional devices. Here, we demonstrate the first non-Abelian holonomic
high-dimensional unitary matrices on multilayer silicon nitride integrated
platform. By leveraging the advantage of integrated platform and geometry
phase, ultracompact footprint, highest order (up to six) and broadband
operation (larger than 100nm) non-Abelian holonomy unitary matrices are
experimentally realized. Our work paves the way for versatile non-Abelian
optical computing devices in integrated photonics.
