Reconfigurable non-Abelian geometric phase in hybrid integrated photonics

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.