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Universal Quantum Computation with Hybrid Spin-Majorana Qubits

We theoretically propose a set of universal quantum gates acting on a hybrid qubit formed by coupling a quantum dot spin qubit and Majorana fermion qubit. First, we consider a quantum dot tunnel-coupled to two topological superconductors. The effective spin-Majorana exchange facilitates a hybrid CNOT gate for which either qubit can be the control or target. The second setup is a modular scalable network of topological superconductors and quantum dots. As a result of the exchange interaction between adjacent spin qubits, a CNOT gate is implemented that acts on neighboring Majorana qubits, and eliminates the necessity of inter-qubit braiding. In both setups the spin-Majorana exchange interaction allows for a phase gate, acting on either the spin or the Majorana qubit, and for a SWAP or hybrid SWAP gate which is sufficient for universal quantum computation without projective measurements.