Majorana-XYZ subsystem code

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.

Figures (6)

• Figure 1: The Majorana-XYZ code. The physical system is a honeycomb arrangement of Majorana fermions, for example, quantised vortices in an $s$-wave superfluid in the topological phase, arranged in place by optical tweezers. Top: the representation in terms of an equivalent spin-$\frac{1}{2}$ model on a triangular lattice. The reference site convention of the Hamiltonian triangles in red, and the four-Majorana interaction terms mapping to the Hamiltonian down and up triangles in blue and magenta respectively. The system is highly frustrated, where the origin of frustration comes from the anti-commutation of any two up-up or down-down corner-sharing Hamiltonian triangles.
• Figure 2: Single-loop operators $\Xi_i^Z$, $\Xi_i^X$, and $\Xi_i^Y$ that are conserved by the Majorana-XYZ Hamiltonian. The dashed site belongs to the next unit cell. The unit cell here is $3\times 3$.
• Figure 3: The bare logical operators satisfying the spin-$1/2$ algebra for one logical qubit. Left: $X^{(\mathrm{L})}$. Mid: $Y^{(\mathrm{L})}$. Right: $Z^{(\mathrm{L})}$. The bare logical operators commute with all the double loops $\Xi_i^A \Xi_{i+1}^A$ and all triangles. In the rest of the paper we do not explicitly visualise the scalar phase factors.
• Figure 4: Three logical qubits (blue triangles). The (dressed) logical operators are shown for one qubit. The dressing is shown by orange triangles.
• Figure 5: Three undetectable errors of weight four. All such errors belong to the gauge group (products of triangle operators) and commute with all stabilisers.