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We propose a topological qubit in which braiding and readout are mediated bythe $4\pi$ Majorana-Josephson effect. The braidonium device consists of threeMajorana nanowires that come together to make a tri-junction; in order tocontrol the superconducting phase differences at the tri-junction the nanowiresare enclosed in a ring made of a conventional superconductor; and in order toperform initialization/readout one of the nanowires is coupled to a fluxoniumqubit through a topological Josephson junction. We analyze how flux-basedcontrol and readout protocols can be used to demonstrate braiding and qubitoperation for realistic materials and circuit parameters.

Braidonium: a braiding quantum circuit based on the $4\pi$ Josephson effect