Controlled-Z Gates via Floquet Coupling to Two Auxiliary States in Transmon Circuits

High-fidelity entangling gates are essential for quantum information processing. In transmon circuits, higher excited states provide additional pathways for engineering interactions within the computational subspace. Here we demonstrate controlled-Z (CZ) gates that exploit two noncomputational states, and, through selective Floquet coupling. The nontrivial parametric modulation can activate both auxiliary transitions from while suppressing the leading exchange between and. Based on this approach, we develop two gate protocols: a nonadiabatic round trip through a bright auxiliary superposition and adiabatic evolution along a zero-energy dark state, both achieving average gate fidelities above 99.99% in numerical simulations. Control of the modulation phase further extends the bright-state protocol to conventional and unconventional geometric CZ gates. Effective-model simulations show that unconventional control improves tolerance to residual ZZ coupling and static qubit-frequency offsets relative to conventional control over the parameter ranges examined. Simulations including neighboring spectator qubits show that the original nonadiabatic bright-state and adiabatic dark-state CZ protocols retain high gate fidelities. This framework offers new routes toward the experimental realization of CZ gates and their geometric implementations.