High-dimensional non-Abelian holonomy in integrated photonics

Non-Abelian holonomy is known for the robust holonomic unitary behavior exhibited. The associated non-Abelian geometric phase is a promising approach for implementing topologically protected computation. But its realization in application-abundant platforms has been largely elusive. In particular, the observation of universal high-order matrices is difficult due to challenges from increasing the dimensions of degenerate subspace. Here we realize a high-dimensional non-Abelian holonomic device on an integrated multilayer silicon nitride platform, which is compatible with the complementary-metal-oxide-semiconductor process. High dimensional (up to 6), broadband (> 100 nm operating bandwidth), and ultra-compact volume non-Abelian holonomy unitary matrices of arbitrary special orthogonal group are observed, and M × N linear holonomic computation architecture is experimentally realized through singular value decomposition. Our work provides a paradigm for versatile applications of non-Abelian geometric phase for both classical and quantum realms.