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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.
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