Long-range coupling of spin qubits in superconductor-semiconductor hybrid structures

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The aim of the project is to theoretically investigate the prospects of using superconducting sheets or wires as couplers that allows for controllably inducing exchange coupling between spin qubits over a distance of up to several microns. Such a coupling is crucial to make spin gubits a scalable architecture. The coupling proceeds via a virtual exchange of a quasiparticle with definite spin through the superconductor. We will investigate the role of magnetic and non-magnetic disorder, spin-orbit interaction, and non-equilibrium quasiparticles and how they influence the gate fidelities, when the gates are mediated by the long-range coupling through the superconductor. In particular, we will study in detail how quasiparticles are generated in the process of performing the gates, which will enable us to find limits on the speed at which the gates can be operated. We will also model the tunnel coupling between the quantum dot and the electron reservoir of the superconductor in detail, as it is the single most-important number determining the strength of the coupling we can hope to achieve. The methods employed are a combination of analytical as well as numerical approaches: e.g. for the clean superconductors, we will solve the Bogoliubov-de Gennes equation. The study of the disordered system is performed by employing semiclassical Green's functions techniques valid for mean free paths much longer than the Fermi wave-length, supplemented by numerical approaches using recursive Green's function in the regime of smaller mean free path. The non-equilibrium problems will be investigated with master equations for the density matrix of the spin qubit that enable us to take both the coherent exchange coupling as well as the different decoherence mechanisms into account.