Majorana-XYZ code[1]

Description

Two-dimensional non-CSS subsystem qubit stabilizer code on a triangular lattice whose gauge group is generated by geometrically local weight-three triangle operators. It encodes one logical qubit with distance proportional to the linear lattice size, while taking only the double loops as stabilizers instead yields a distance-three subspace code whose rate tends to one. The stabilizer generators are non-local loop products, but their syndrome can in principle be inferred from the local gauge measurements, making the code suitable for architectures with native few-qubit measurements.

The gauge generators are the terms of a frustrated three-spin Hamiltonian, with every up- and down-pointing triangle of the lattice hosting a weight-three Pauli operator that assigns \(Z\), \(X\), and \(Y\) to its three vertices in an orientation-dependent pattern, \begin{align} T_{k}^{\nabla} = Z_k X_i Y_j \qquad\text{and}\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, so the gauge group is non-Abelian and the associated Hamiltonian is strongly frustrated.

For each Pauli type \(A \in \{X,Y,Z\}\) there is one lattice direction along which the product of \(A\) over a straight line winding around the torus commutes with every triangle operator; there are \(3L\) such weight-\(L\) loop operators \(\Xi^{A}_{l}\). 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 and are products of triangle operators; they generate the stabilizer group. The center has \(|\mathsf{S}| = 3L-3\) independent generators for every \(L\) — double loops of weight \(2L\), together with one additional non-double-loop generator when \(L\) is even — leaving \(g = (|\mathsf{G}|-|\mathsf{S}|)/2 = L^2-3L+2\) gauge qubits and a single logical qubit. The reported subsystem parameters are \([[L^2,1,L^2-3L+2,L]]\) [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]. The double loops are no longer all independent for even \(L\), where one additional non-double-loop generator enters the center of the gauge group. For even \(L\) this code therefore has one logical qubit more than the \(k+g\) of the subsystem code, since that extra center generator is released as well.