The current-mirror circuit [A. Kitaev, arXiv:cond-mat/0609441 (2006)] exhibits a robust ground-state degeneracy and wave functions with disjoint support for appropriate circuit parameters. In this protected regime, Cooper-pair excitons form the relevant low-energy excitations. Based on a full circuit analysis of the current-mirror device, we introduce an effective model that systematically captures the relevant low-energy degrees of freedom, and is amenable to diagonalization using Density Matrix Renormalization Group (DMRG) methods. We find excellent agreement between DMRG and exact diagonalization, and can push DMRG simulations to much larger circuit sizes than feasible for exact diagonalization. We discuss the spectral properties of the current-mirror circuit, and predict coherence times exceeding 1 ms in parameter regimes believed to be within reach of experiments.
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