  Time-periodic systems allow engineering new effective Hamiltonians from
limited physical interactions. For example, the inverted position of the
Kapitza pendulum emerges as a stable equilibrium with rapid drive of its pivot
point. In this work, we propose the $\textit{Kapitzonium}$: a Floquet qubit
that is the superconducting circuit analog of a mechanical Kapitza pendulum.
Under periodic driving, the emerging qubit states are exponentially protected
against bit and phase flips caused by dissipation, which is the primary source
of decoherence of current qubits. However, we find that dissipation causes
leakage out of the Floquet qubit subspace. We engineer a passive cooling scheme
to stabilize the qubit subspace, which is crucial for high fidelity quantum
control under dissipation. Furthermore, we introduce a hardware-efficient
fluorescence-based method for qubit measurement and discuss the experimental
implementation of the Floquet qubit. The proposed Kapitzonium is one of the
simplest Floquet qubits that can be realized with current technology -- and it
already has many intriguing features and capabilities. Our work provides the
first steps to develop more complex Floquet quantum systems from the ground up
to realize large-scale protected engineered dynamics.
