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, $\ket{02}$ and $\ket{20}$, through selective Floquet coupling. The nontrivial parametric modulation can activate both auxiliary transitions from $\ket{11}$ while suppressing the leading exchange between $\ket{01}$ and $\ket{10}$. 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.