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, so that a rectangular patch of height \(d\) and width \(2d\) holding one domain wall encodes three logical qubits [1]. Each twist defect at an end of the domain wall is a new endpoint for logical operators, which flip between \(X\) and \(Z\) type across the wall, so that loops and mixed Pauli strings become additional independent logical operators. Merging the boundaries of an \(n_{\text{row}}\times m_{\text{col}}\) grid of such rectangles gives one patch encoding \(3 n_{\text{row}} m_{\text{col}}\) logical qubits, approaching one logical qubit per \(d(d-1)\) physical qubits, roughly twice the rate of rotated surface-code patches [1].
Twist defects carry non-CSS weight-five stabilizer generators containing a \(Y\) Pauli operator, and the packing is made practical by a syndrome-extraction cycle for them that keeps the code decodable by minimum-weight perfect matching. Rather than measuring such a generator directly, the cycle 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].
One rectangular patch with two twist defects has an end-cycle stabilizer group forming a \([[49,3,5]]\) code [1; Fig. 10]. Concatenating each column of the dense packing with an \([[n,n-2,2]]\) error-detecting code roughly doubles the distance [1].
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. 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. 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 observed under circuit-level noise 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 uniform depolarizing circuit-level noise 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 [1]. The corresponding rate for an idling rotated surface-code patch is \(4^{-d}/10\) [1].Twist defects can be walked one lattice spacing every two rounds. The walk reverses the controls and targets of the two-qubit gates and alternates 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. This roughly doubles the distance [1]. The inner stabilizer generators are \(X^{\otimes n}\) and \(Z^{\otimes n}\). 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. They 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. This roughly doubles the distance [1]. The inner stabilizer generators are \(X^{\otimes n}\) and \(Z^{\otimes n}\). 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. They 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. This roughly doubles the distance [1]. The inner stabilizer generators are \(X^{\otimes n}\) and \(Z^{\otimes n}\). 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. They 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 (2026-09-26) — most recent
- Victor V. Albert (2026-08-26)
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