Abstract
We introduce an exactly solvable model of interacting Majorana fermions realizing $Z_{2}$ topological order with a $Z_{2}$ fermion parity grading and lattice symmetries permuting the three fundamental anyon types. We propose a concrete physical realization by utilizing quantum phase slips in an array of Josephson-coupled mesoscopic topological superconductors, which can be implemented in a wide range of solid state systems, including topological insulators, nanowires or two-dimensional electron gases, proximitized by $s$-wave superconductors. Our model finds a natural application as a Majorana fermion surface code for universal quantum computation, with a single-step stabilizer measurement requiring no physical ancilla qubits, increased error tolerance, and simpler logical gates than a surface code with bosonic physical qubits. We thoroughly discuss protocols for stabilizer measurements, encoding and manipulating logical qubits, and gate implementations.
Highlights
As originally proposed by Ettore Majorana, the Majorana fermion is a particle that is its own antiparticle [1]
We introduce a new scheme for surface code quantum computation that uses Majorana fermions as the fundamental physical degrees of freedom and exploits their unique properties for encoding and manipulating logical qubits
We describe a detailed physical implementation of the “Majorana fermion surface code,” including physical qubit and stabilizer measurements, the creation of logical qubits, error correction, and logical gate operations required for universal quantum computation
Summary
As originally proposed by Ettore Majorana, the Majorana fermion is a particle that is its own antiparticle [1]. The “surface code” [24,25] provides an alternative approach to universal quantum computation that uses measurements in an Abelian topological phase for gate operations and error correction. We demonstrate that charging energyinduced quantum phase slips in superconducting arrays with Majorana fermions generates the required multifermion plaquette interactions, providing a physical realization of our model. We describe a detailed physical implementation of the “Majorana fermion surface code,” including physical qubit and stabilizer measurements, the creation of logical qubits, error correction, and logical gate operations required for universal quantum computation. We propose a physical realization of this model that uses the charging energy in an array of mesoscopic superconductors [18] to implement the required nonlocal interactions between multiple Majorana fermions. We present a physical implementation of the Majorana surface code and propose detailed protocols for performing gate operations for universal quantum computation
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