  Properties of time-periodic Hamiltonians can be exploited to increase the
dephasing time of qubits and to design protected one and two-qubit gates.
Recently, Huang et al. [Phys. Rev. Applied 15, 034065 (2021)] have shown that
time-dependent Floquet states offer a manifold of working points with dynamical
protection larger than the few usual static sweet spots. Here, we use the
framework of many-mode Floquet theory to describe approaches to robustly
control Floquet qubits in the presence of multiple drive tones. Following the
same approach, we introduce a longitudinal readout protocol to measure the
Floquet qubit without the need of first adiabatically mapping back the Floquet
states to the static qubit states, which results in a significant speedup in
the measurement time of the Floquet qubit. The analytical approach developed
here can be applied to any Hamiltonian involving a small number of distinct
drive tones, typically the study of standard parametric gates for qubits
outside of the rotating-wave approximation.
