  Superconducting circuits rank among the most interesting architectures for
the implementation of quantum information processing devices. The recently
proposed 0-$\pi$ qubit [Brooks et al., Phys. Rev. A ${\bf 87}$, 52306 (2013)]
promises increased protection from spontaneous relaxation and dephasing. In
practice, this ideal behavior is only realized if the parameter dispersion
among nominally identical circuit elements vanishes. In this paper we present a
theoretical study of the more realistic scenario of slight variations in
circuit elements. We discuss how the coupling to a spurious, low-energy mode
affects the coherence properties of the 0-$\pi$ device, investigate the
relevant decoherence channels, and present estimates for achievable coherence
times in multiple parameter regimes.
