  Superconducting qubit designs vary in complexity from single- and
few-junction systems, such as the transmon and flux qubits, to the
many-junction fluxonium. Here we consider the question of wether the many
degrees of freedom in the fluxonium circuit can limit the qubit coherence time.
Such a limitation is in principle possible, due to the interactions between the
low-energy, highly anharmonic qubit mode and the higher-energy, weakly
anharmonic collective modes. We show that so long as the coupling of the
collective modes with the external electromagnetic environment is sufficiently
weaker than the qubit-environment coupling, the qubit dephasing induced by the
collective modes does not significantly contribute to decoherence. Therefore,
the increased complexity of the fluxonium qubit does not constitute by itself a
major obstacle for its use in quantum computation architectures.
