Tetron subsystem code[1]

Description

Member of a family of Majorana subsystem stabilizer codes on \(n\) tetrons obtained from an \([[n,k_b,d_b]]\) qubit stabilizer code and an \([n,k,d_c]\) classical binary code. The qubit code’s Pauli operators become weight-two Majorana operators on the tetrons, and the classical code’s parity checks become stabilizers that are products of four-Majorana tetron parities. A tetron parity flips only under an odd-weight (fermionic) error, so these checks detect such errors, while tetron parities not fixed by them are gauge operators rather than further stabilizer generators [1].

A tetron is a superconducting island hosting four Majorana zero modes \(\gamma_a,\gamma_b,\gamma_c,\gamma_d\), on which only parities of two modes are measured, and it 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 using only two-mode parity measurements [1].

The Pauli operators of the qubit code are mapped onto the Majorana operators of the set \(\mathcal{R}\), one tetron per qubit. Each row of the classical parity-check matrix \(H\) contributes a stabilizer equal to the product of the tetron parities on its support. A tetron parity cannot be measured directly. Such a stabilizer is instead realized as the product of an existing stabilizer supported on those tetrons and a copy of it with the operators on those tetrons 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 parity 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]. Taking the classical code to have no logical bits places every tetron parity in the stabilizer group and leaves no gauge qubits. This recovers concatenation with the tetron code [2][3; Lemma 1][4; Sec. IV]. Relative to that earlier non-subsystem construction, the subsystem code corrects both odd- and even-weight errors with fewer stabilizer generators [1].

The \([[10,1,2,3]]\) code is obtained from the five-qubit perfect code and the \([5,2,3]\) classical code [1; Sec. 3.5]. See Ref. [1; Table 1] for the instances obtained from the \([[6,1,3]]\) code with the \([6,1,6]\) and \([6,3,3]\) codes and from the Steane code.