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Initializing the ground state of a quantum bit (qubit) based on Majorana zero modes is one of the most pressing issues for future topological quantum computers. We explore a parity-to-magnetism protocol for initializing such topological qubits based on magnet-superconductor hybrid networks by coupling magnetic chains to a single molecule magnet. The coupling can be activated by switching the spin state of the molecule, allowing the ground state parity of the chain to be controlled. We demonstrate that initialization with either parity for a Majorana qubit can be achieved. We then introduce the dressed Majorana qubit, which includes the state of the molecule in the definition of the logical qubit. Using this definition we can initialize a qubit without high-precision timing., Comment: 6 pages, 4 figures

Qubits built out of Majorana zero modes have long been theorized as a potential pathway toward fault-tolerant topological quantum computation. Almost unavoidable in these processes is Majorana wavefunction overlap, known as hybridization, which arise throughout the process when Majorana modes get close to each other. This breaks the ground state degeneracy, leading to qubit errors in the braiding process. This work presents an accessible method to track transitions within the low-energy subspace and predict the output of braids with hybridized Majorana modes. As an application, we characterize Pauli qubit-errors, as demonstrated on an X-gate, critical for the successful operation of any quantum computer. Further, we perform numerical simulations to demonstrate how to utilize the hybridization to implement arbitrary rotations, along with a two-qubit controlled magic gate, thus providing a demonstration of universal quantum computing.

Qubits built out of Majorana zero modes (MZMs) constitute the primary path towards topologically protected quantum computing. Simulating the braiding process of multiple MZMs corresponds to the quantum dynamics of a superconducting many-body system. It is crucial to study the Majorana dynamics both in the presence of all other quasiparticles and for reasonably large system sizes. We present a method to calculate arbitrary many-body wavefunctions as well as their expectation values, correlators and overlaps from time evolved single-particle states of a superconductor, allowing for significantly larger system sizes. We calculate the fidelity, transition probabilities, and joint parities of Majorana pairs to track the quality of the braiding process. We show how the braiding success depends on the speed of the braid. Moreover, we demonstrate the topological CNOT two-qubit gate as an example of two-qubit entanglement. Our work opens the path to test and analyze the many theoretical implementations of Majorana qubits. Moreover, this method can be used to study the dynamics of any non-interacting superconductor., Comment: v2: revised version, including the demonstration of CNOT gate via braiding