Visibility of a topological qubit quantum dot measurement

Topological superconductor-quantum dot-topological superconductor (TS-QD-TS) system is of current interest for topological quantum computing, being a proposed platform for detecting Majorana zero modes (MZMs) and for measuring a topological qubit state. Here, using perturbation theory we derive expression for Josephson energy in this system, with a particular emphasis on a term associated with a coupling of the QD to quasiparticle continuums in each TS, arising in the fourth order. Furthermore, we study how 1/f flux noise, typical for SQUID-type devices, interferes measurement visibility of the qubit. We find that even though coupling of the QD to quasiparticle continuums does not directly affect measurement outcomes, the flux noise might considerably decrease the signal to noise ratio. We calculate experimental parameter values for which the effect is minimized and a best signal to noise ratio can be achieved.