Dense twist-defect surface code[1]
Description
Member of a family of twist-defect surface codes whose twist defects are packed as densely as circuit-level error mechanisms allow, roughly doubling the number of logical qubits per physical qubit relative to rotated surface-code patches. The packing is made practical by a syndrome-extraction cycle for the weight-five twist-defect stabilizer generators that needs only degree-three qubit connectivity, six layers of operations, and decoding by minimum-weight perfect matching. Twist positions are chosen against measured circuit-level error rates rather than against the code distance.
Twist defects sit at the endpoints of a domain wall and carry non-CSS weight-five stabilizer generators containing a \(Y\) Pauli operator. Rather than measuring such a generator directly, the construction measures lower-weight gauge operators that commute with the stabilizer group but anticommute with each other. Individual gauge outcomes are random, while the parity of two of them reconstructs the twist-defect stabilizer generator. An additional weight-two domain wall removes matchable interior boundaries. The resulting cycle uses one reset layer, four layers of two-qubit gates, and one measurement layer, on a degree-three connectivity graph. It loses no distance parallel to the domain wall and one unit perpendicular to it, or no distance at all in an eight-layer variant. Earlier twist-defect circuits needed at least seven layers and degree-six connectivity [2,3], or halved the distance along the domain wall and gave up matching-based decoding [4].
Three packings are given. A rectangular patch with two twist defects encodes three logical qubits in the footprint of two distance-\(d\) rotated surface-code patches, and one example has an end-cycle stabilizer group forming a \([[49,3,5]]\) code. Merging the boundaries of a grid of \(n_{\text{row}}\times m_{\text{col}}\) such rectangles gives a single patch encoding \(3 n_{\text{row}} m_{\text{col}}\) logical qubits, approaching one logical qubit per \(d(d-1)\) physical qubits. Concatenating each column of that packing with an \([[n,n-2,2]]\) error-detecting code roughly doubles the distance.
Protection
Placement of the twist defects is governed by two competing error mechanisms. Logical \(X\) and \(Z\) errors propagate diagonally, with a distance set by the \(L_\infty\) norm, whereas logical \(Y\) errors propagate horizontally or vertically, with a distance set by the \(L_1\) norm. Maintaining the distance therefore requires placing twist defects far enough from each other and from the boundaries of the patch [1].Decoding
Correlated minimum-weight perfect matching via sparse blossom [5], applied to detector error models constructed with Stim [6]. The gauge-operator measurement scheme is designed to keep the code matchable, unlike lower-depth twist implementations that forfeit polynomial-time decoding [4].Fault Tolerance
Diagonally propagating \(YY\) hook errors would halve the circuit-level distance, but their probability is suppressed because they require one specific low-entropy configuration of two-qubit depolarizing faults, each of small probability [1]. Twist defects are therefore placed according to the larger effective distance seen in simulation rather than the graphlike distance. Neither the code-level nor the circuit-level distance is an upper or a lower bound on the performance of these codes.Under a uniform depolarizing circuit-level noise model at physical error rate \(10^{-3}\), the per-logical per-round logical error rate is \(3.5^{-d}/10\) for both the three-qubit rectangular patch and the dense packing, against \(4^{-d}/10\) for an idling rotated surface-code patch [1].Twist defects can be walked one lattice spacing every two rounds by reversing the controls and targets of the two-qubit gates and alternating with a reflection, extending the walking technique for surface-code patches [7].Cousins
- Yoked 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 [1]. 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.
- Concatenated qubit 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 [1]. 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.
- \([[2m,2m-2,2]]\) error-detecting 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 [1]. 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.
Primary Hierarchy
References
- [1]
- G. H. Low, W. J. Huggins, D. W. Berry, T. Khattar, A. F. White, N. C. Rubin, and R. Babbush, “A Denser Planar Surface Code”, (2026) arXiv:2605.30455
- [2]
- D. Litinski and F. von Oppen, “Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes”, Quantum 2, 62 (2018) arXiv:1709.02318 DOI
- [3]
- C. Chamberland and E. T. Campbell, “Circuit-level protocol and analysis for twist-based lattice surgery”, Physical Review Research 4, (2022) arXiv:2201.05678 DOI
- [4]
- G. P. Gehér, O. Crawford, and E. T. Campbell, “Tangling Schedules Eases Hardware Connectivity Requirements for Quantum Error Correction”, PRX Quantum 5, (2024) arXiv:2307.10147 DOI
- [5]
- O. Higgott and C. Gidney, “Sparse Blossom: correcting a million errors per core second with minimum-weight matching”, Quantum 9, 1600 (2025) arXiv:2303.15933 DOI
- [6]
- C. Gidney, “Stim: a fast stabilizer circuit simulator”, Quantum 5, 497 (2021) arXiv:2103.02202 DOI
- [7]
- M. McEwen, D. Bacon, and C. Gidney, “Relaxing Hardware Requirements for Surface Code Circuits using Time-dynamics”, Quantum 7, 1172 (2023) arXiv:2302.02192 DOI
Page edit log
- Victor V. Albert (2026-08-26) — most recent
Cite as:
“Dense twist-defect surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/dense_twist_defect, arXiv:2606.11484