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We propose a topological qubit in which braiding and readout are mediated by the $4pi$ Majorana-Josephson effect. The braidonium device consists of three Majorana nanowires that come together to make a tri-junction; in order to control the superconducting phase differences at the tri-junction the nanowires are enclosed in a ring made of a conventional superconductor; and in order to perform initialization/readout one of the nanowires is coupled to a fluxonium qubit through a topological Josephson junction. We analyze how flux-based control and readout protocols can be used to demonstrate braiding and qubit operation for realistic materials and circuit parameters.
The $4pi$-periodic Josephson effect is an indicator of Majorana zero modes and a ground-state degeneracy which are central to topological quantum computation. However, the observability of a $4pi$-periodic Josephson current-phase relation (CPR) is hi
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