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Holonomic quantum computing (HQC) functions by transporting an adiabaticallydegenerate manifold of computational states around a closed loop in acontrol-parameter space; this cyclic evolution results in a non-Abeliangeometric phase which may couple states within the manifold. Realizing therequired degeneracy is challenging, and typically requires auxiliary levels orintermediate-level couplings. One potential way to circumvent this is throughFloquet engineering, where the periodic driving of a nondegenerate Hamiltonianleads to degenerate Floquet bands, and subsequently non-Abelian gaugestructures may emerge. Here we present an experiment in ultracold $^{87}$Rbatoms where atomic spin states are dressed by modulated RF fields to induceperiodic driving of a family of Hamiltonians linked through a fully tuneableparameter space. The adiabatic motion through this parameter space leads to theholonomic evolution of the degenerate spin states in $SU(2)$, characterized bya non-Abelian connection. We study the holonomic transformations of spineigenstates in the presence of a background magnetic field, characterizing thefidelity of these gate operations. Results indicate that while the Floquetengineering technique removes the need for explicit degeneracies, it inheritsmany of the same limitations present in degenerate systems.

Demonstration of Floquet engineered non-Abelian geometric phase for holonomic quantum computing