Majorana-XYZ subsystem code

TL;DR

The paper introduces the Majorana-XYZ code, a topologically protected quantum error correction code with macroscopic logical qubits, derived from Majorana fermions on a honeycomb lattice, featuring 3-local measurements and high error detection capabilities.

Contribution

It presents a novel Majorana-XYZ subsystem code combining topological and gauge code features with scalable logical qubits and practical measurement operations.

Findings

Detects all 1- and 2-qubit errors

Confines undetected high-weight errors to gauge group

Supports macroscopic logical qubits with 3-local measurements

Abstract

We present a new type of a quantum error correction code, termed Majorana-XYZ code, where the logical quantum information scales macroscopically yet is protected by topologically non-trivial degrees of freedom. It is a $[n,k,g,d]$ subsystem code with $n=L^2$ physical qubits, $k= \lfloor L/2 \rfloor$ logical qubits, $g \sim L^2$ gauge qubits, and distance $d = L$. The physical check operations, i.e. the measurements needed to obtain the error syndrome, are $3$-local and nearest-neighbour. The code detects every 1- and 2-qubit error, and every error of weight 3 and higher (constrained by the distance) that is not a product of the 3-qubit check operations, however, these products act only on the gauge qubits leaving the code space invariant. The undetected weight-3 and higher operators are confined to the gauge group and do not affect logical information. While the code does not have local stabiliser generators, the logical qubits cannot be modified locally by an undetectable error, and in this sense the Majorana-XYZ code combines notions of both topological and local gauge codes while providing a macroscopic number of topological logical qubits. Taken as a non-gauge stabiliser code we can encode $k \sim L^2 - 3L$ logical qubits into $L^2$ physical qubits; however, the check operators then become weight $2L$. The code is derived from an experimentally promising system of Majorana fermions on the honeycomb lattice with only nearest-neighbour interactions.