Encoding Majorana codes*

A Majorana fermion a type of particle which is its own antiparticle. They can emerge as quasiparticles which are bound to defects in superconducting systems, and exist at zero energies, hence they are also known as Majorana zero modes or MZM's. There are many theoretical predictions, as well as experimental developments in trying to find a Majorana fermion, but to date, the Majorana remains elusive. Majoranas are also interesting from a quantum information perspective because information can be encoded into non local degrees of freedom (such as into the edge states of a 1D Kitaev chain)  and because of their non-Abelian exchange statistics, which allows one to do error robust gates in the form of braiding operations. Motivated by their potential, Majorana fermionic codes were initially introduced by Bravyi et al as extensions of the Kitaev chain, and integrates Majorana fermions into an error correcting code where qubits are encoded into the logical subspace of physical Majoranas, and thus these codes can protect against low weight fermionic noise. A distinct advantage of fermion based codes is that they admit a wider range of codes, since not all fermion codes can be mapped to qubit stabilizer codes for the same parameters. 

In order to run an error correction protocol on Majorana fermion codes, we first need a method to encode logical qubits in terms of physical fermions. In this paper, we use the so-called logical braid gates to construct an algorithm that takes as input a extbf{general} Majorana stabilizer code, and gives an encoding circuit, in terms of quadratic and quartic Majorana braid gates. We provide two approaches, one that uses an ancilla mode and works for any stabilizer code, while the second approach does not use an ancilla but does not work when the total parity is inside the stabilizer group.  

We hope this result will be useful in Majorana based platforms that use braiding operations to perform gates on Majorana modes.