  Breaking the error-threshold would mark a milestone in establishing quantum
advantage for a wide range of relevant problems. One possible route is to
encode information redundantly in a logical qubit by combining several noisy
qubits, providing an increased robustness against external perturbations. We
propose a setup for a logical qubit built from superconducting qubits (SCQs)
coupled to a microwave cavity-mode. Our design is based on a recently
discovered geometric stabilizing mechanism in the Bose-Hubbard wheel (BHW),
which manifests as energetically well-separated clusters of many-body
eigenstates. We investigate the impact of experimentally relevant perturbations
between SCQs and the cavity on the spectral properties of the BHW. We show that
even in the presence of typical fabrication uncertainties, the occurrence and
separation of clustered many-body eigenstates is extremely robust. Introducing
an additional, frequency-detuned SCQ coupled to the cavity yields duplicates of
these clusters, that can be split up by an on-site potential. We show that this
allows to (i) redundantly encode two logical qubit states that can be switched
and read out efficiently and (ii) can be separated from the remaining many-body
spectrum via geometric stabilization. We demonstrate at the example of an
X-gate that the proposed logical qubit reaches single qubit-gate fidelities
$>0.999$ in experimentally feasible temperature regimes
$\sim10-20\,\mathrm{mK}$.
