Majorana-XYZ code [1]

Description

Non-CSS subsystem qubit stabilizer code on a periodic \(L\times L\) triangular lattice whose gauge group is generated by weight-three triangle operators, each placing \(Z\), \(X\), and \(Y\) on the corners of one plaquette. For each Pauli type, straight-line loops along one lattice direction commute with every triangle operator, and the product of two adjacent parallel loops is itself a product of triangle operators. Such double loops are stabilizers and single loops are the bare logical operators, so the code encodes one logical qubit with distance \(L\) using only three-qubit checks [1].

The triangle operators are the terms of a frustrated three-spin Hamiltonian. Every up- and down-pointing triangle of the lattice hosts one such operator. It assigns \(Z\), \(X\), and \(Y\) to the three vertices in an orientation-dependent pattern, \begin{align} T_{k}^{\nabla} = Z_k X_i Y_j \qquadand\qquad T_{j}^{\Delta} = Z_j Y_k X_l~,\tag*{$(1)$} \end{align}where the site labels follow the two triangle orientations [1; Fig. 1]. The two orientations mutually commute, but any two corner-sharing triangles of the same orientation anticommute. The gauge group is therefore non-Abelian, and the Hamiltonian is strongly frustrated. The \(3L\) weight-\(L\) loops that commute with every triangle operator are the loop operators \(\Xi^{A}_{l}\), one for each Pauli type \(A \in \{X,Y,Z\}\) and line position \(l\) [1; Fig. 2]. Products \(\Xi^{A}_{i}\Xi^{A}_{i+1}\) of two adjacent parallel loops of the same type intersect any other loop an even number of times. The stabilizer group, i.e., the center of the gauge group, has \(|\mathsf{S}| = 3L-3\) independent generators for every \(L\). These are nonlocal double loops of weight \(2L\), together with one additional non-double-loop generator when \(L\) is even. This leaves \(g = (|\mathsf{G}|-|\mathsf{S}|)/2 = L^2-3L+2\) gauge qubits and a single logical qubit, for subsystem code parameters \([[L^2,1,L^2-3L+2,L]]\) [1]. The physical checks are the triangle operators rather than the loops, and each double-loop syndrome bit is the product of the outcomes of the triangle operators composing it [1].

Taking the double loops alone as the stabilizer group of a subspace code, without designating any gauge qubits, yields an \([[L^2,L^2-3L+3+((L+1) \bmod 2),3]]\) code whose rate tends to one [1; Sec. II.A]. For even \(L\), the double loops are not all independent, and the additional center generator of the subsystem code is released as well. This code therefore has one logical qubit more than the \(k+g\) of the subsystem code when \(L\) is even.