Tetron subsystem code[1]

Description

Member of a family of Majorana subsystem stabilizer codes on tetrons, obtained from a qubit stabilizer code together with a classical binary code. Tetron parity operators are placed in the gauge group rather than the stabilizer group. This allows the code to correct both odd- and even-weight Majorana errors while using fewer stabilizer generators than the earlier non-subsystem tetron construction with the same capability [1].

A tetron is a superconducting island hosting four Majorana zero modes \(\gamma_a,\gamma_b,\gamma_c,\gamma_d\), and stores a qubit in the form of operator parity. Its six weight-two Majorana operators split into two sets, \begin{align} \mathcal{R} &= \{\gamma_b\gamma_c,~\gamma_a\gamma_c,~\gamma_a\gamma_b\}\tag*{(1)}\\ \mathcal{R}^{\prime} &= \{\gamma_a\gamma_d,~\gamma_d\gamma_b,~\gamma_c\gamma_d\}~, \tag*{(2)}\end{align} identified with the Pauli operators \(X,Y,Z\) and their primed counterparts, respectively [1; Fig. 1]. Corresponding operators of the two sets return the same parity under an even-weight error and opposite parities under an odd-weight error. Comparing them therefore reveals whether an error has odd weight, and requires no measurement beyond the conventional two-Majorana-per-tetron parity measurements [1].

The construction takes a non-subsystem \([[n,k_b,d_b]]\) qubit stabilizer code together with an \([n,k,d_c]\) classical binary code. One tetron is assigned to each qubit, and the Pauli operators of the qubit code are mapped onto the Majorana operators of the set \(\mathcal{R}\). Each row of the classical parity-check matrix \(H\) contributes a stabilizer equal to the product of the tetron parity operators on its support. Since a tetron parity cannot be measured directly, such a stabilizer is realized as the product of an existing stabilizer supported on those tetrons and a modified copy of it in which the operators on those tetrons are switched from \(\mathcal{R}\) to \(\mathcal{R}^{\prime}\). The two codes must therefore be compatible: every row of \(H\) has to admit a stabilizer of the qubit code supported on at least the tetrons in that row [1].

Each row of the systematic generator matrix \(G=[I_k|P]\) contributes two gauge operators: the tetron operator at the row’s first nonzero entry, and the \(\gamma_d\)-string supported on all of its nonzero entries. These \(2k\) operators generate a gauge group of order \(2^{2k}\), yielding \(k\) gauge qubits. The output is a \([[2n,k_b,k,d_f]]\) Majorana subsystem code on \(n\) tetrons, i.e., \(2n\) fermionic modes, with \(n-k_b\) stabilizers inherited from the qubit code and \(n-k\) from the classical code [1]. The classical code thus interpolates between two familiar constructions. Taking it to have no logical bits places every tetron parity in the stabilizer group and leaves no gauge qubits, recovering concatenation with the tetron code [2][3; Lemma 1][4; Sec. IV].

Reported instances are the \([[10,1,2,3]]\) code from the five-qubit perfect code and the \([5,2,3]\) classical code, the \([[12,1,1,6]]\) and \([[12,1,3,3]]\) codes from the \([[6,1,3]]\) code with the \([6,1,6]\) repetition and \([6,3,3]\) classical codes, and the \([[14,1,4,3]]\) code from the Steane code and the \([7,4,3]\) Hamming code [1].