Yoked surface code[1]

Cousins

• Quantum multi-dimensional parity-check (QMDPC) code— Yoked surface codes are concatenations of QMDPC codes with rotated surface codes.
• Rotated surface code— Yoked surface codes are concatenations of QMDPC codes with rotated surface codes.
• Dense twist-defect surface code— Each column of the twist-defect dense packing is concatenated with an \([[n,n-2,2]]\) error-detecting code as the inner code, in the concatenation convention of the Zoo, which roughly doubles the distance [3]. The inner stabilizer generators are \(X^{\otimes n}\) and \(Z^{\otimes n}\), and two of the \(n\) outer code blocks serve as yoke qubits, so that the \(j\)th encoded qubit can be given the logical basis \(X_j X_{n-1}\) and \(Z_j Z_{n-2}\). This is the twist-defect analogue of the yoked surface code, which uses a column of one-qubit rotated surface-code patches in place of the dense packing. Unlike in that case, the logical operators of the dense packing generally lie in the interior, and must be moved to a boundary by twist-defect lattice surgery before the parity checks can be measured.