Encoding Majorana codes

TL;DR

This paper introduces algorithms to compute unitary encoding circuits for Majorana codes, ensuring fermionic parity preservation, using stabilizer matrices and Gaussian elimination techniques.

Contribution

It presents two novel approaches for encoding Majorana codes, one with ancilla modes applicable to all codes, and another without ancilla for specific cases.

Findings

Algorithm using ancilla works for all Majorana stabilizer codes.

Fermionic Clifford operations replace row operations in Gaussian elimination.

Encoding circuits preserve fermionic parity in Majorana codes.

Abstract

To implement a quantum error correction protocol, we first need a scheme to prepare our state in the correct subspace of the code, and this can be done using a unitary encoding circuit. Majorana codes are special since any gates that transform such codes must preserve fermionic parity. In this paper, we present an algorithm that uses the stabilizer matrix to compute unitary encoding circuits for Majorana codes. We present two approaches, both of which use a version of Gaussian elimination with row operations replaced with elementary fermionic Clifford operations. One approach uses an additional ancilla mode and works for all Majorana stabilizer codes, while the second approach does not use ancilla but does not work if the total parity is inside the stabilizer group.