Floquet-engineered degeneracies for holonomic gates are fragile

Holonomic gates, which rely on non-Abelian geometric phases in degenerate systems, are commonly thought to be robust to various errors. Several recent proposals have introduced a new approach to universal holonomic quantum computing which uses Floquet engineering to obtain the required degeneracy, and this approach seems to promise the same robustness. We demonstrate these gates on the spin states of a rubidium-87 BEC, where fast modulations produce a degeneracy between the spins permitting their holonomic evolution. The presence of external noise perturbs this degeneracy, necessitating a more generalized treatment of the system's evolution which includes dynamical contributions to the phase. We compare the evolution of this system to a more general theory, showing the newly-acquired degeneracy is not robust.