Modeling noise and error correction for Majorana-based quantum computing

Christina Knapp1, Michael Beverland2, Dmitry I. Pikulin3, and Torsten Karzig3

1Department of Physics, University of California, Santa Barbara, California 93106 USA
2Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052 USA
3Station Q, Microsoft Research, Santa Barbara, California 93106-6105 USA

Majorana-based quantum computing seeks to use the non-local nature of Majorana zero modes to store and manipulate quantum information in a topologically protected way. While noise is anticipated to be significantly suppressed in such systems, finite temperature and system size result in residual errors. In this work, we connect the underlying physical error processes in Majorana-based systems to the noise models used in a fault tolerance analysis. Standard qubit-based noise models built from Pauli operators do not capture leading order noise processes arising from quasiparticle poisoning events, thus it is not obvious $\textit{a priori}$ that such noise models can be usefully applied to a Majorana-based system. We develop stochastic Majorana noise models that are generalizations of the standard qubit-based models and connect the error probabilities defining these models to parameters of the physical system. Using these models, we compute pseudo-thresholds for the $d=5$ Bacon-Shor subsystem code. Our results emphasize the importance of correlated errors induced in multi-qubit measurements. Moreover, we find that for sufficiently fast quasiparticle relaxation the errors are well described by Pauli operators. This work bridges the divide between physical errors in Majorana-based quantum computing architectures and the significance of these errors in a quantum error correcting code.

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