Observation of generic U(m) non-Abelian holonomy in photonics

Non-Abelian geometric phases form the foundation of fault-tolerant holonomic quantum computation. An "all-geometric" approach leveraging these phases enables robust unitary operations in condensed matter systems. Photonics, with rich degrees of freedom, offer a highly promising platform for non-Abelian holonomy. Yet, achieving universal unitary transformations in photonic holonomy remain elusive. Intrinsic positive real couplings in dissipationless photonic waveguides restrict holonomy to special orthogonal matrices, falling short of universal quantum gates or arbitrary linear operations. Here, we introduce artificial gauge fields (AGFs) to enable complex-valued couplings, expanding photonic holonomy to the full unitary group. We realize generic U(2) transformations and synthesize higher dimensional U(m) operations (up to U(4)) in integrated photonics. Our results open doors toward the transformative "all-geometric-phase" approach in photonic computing in both classical and quantum realms.