Rotated surface code[14] 

Also known as Checkerboard code, Medial surface code, Rectified surface code.


Variant of the surface code defined on a square lattice that has been rotated 45 degrees such that qubits are on vertices, and both \(X\)- and \(Z\)-type check operators occupy plaquettes in an alternating checkerboard pattern.

Stabilizer generators for this code are shown in Figure I.

Figure I: Stabilizer generators of a 2D rotated surface code with open boundaries. The generators are four-body operators on the corners of squares in the bulk and two-body operators on the boundaries. Red regions correspond to \(X\) operators while blue regions to \(Z\) operators.


The \([[L^2,1,L]]\) variant [1] includes the \([[9,1,3]]\) surface-17 code, named as such because 8 ancilla qubits are used for check operator measurements alongside the 9 physical qubits.


Only certain syndrome extraction schedules are distance-preserving [4].Local neural-network using 3D convolutions, combined with a separate global decoder [5].

Fault Tolerance

A particular choice of CNOT gates during syndrome extraction is required to avoid hook errors and be fault-tolerant to syndrome qubit errors [4,6,7].


  • Kitaev surface code — The lattice of the rotated surface code can be obtained by taking the medial graph of the surface code lattice (treated as a graph) and applying a similar procedure to construct the check operators [1,8][9; Fig. 8]. Applying the quantum Tanner transformation to the surface code yields the rotated surface code [10,11].
  • Quantum Tanner code — Applying the quantum Tanner transformation to the surface code yields the rotated surface code [10,11].
  • Hierarchical code — Hierarchical codes are concatenations of constant-rate QLDPC codes with rotated surface codes.
  • Yoked surface code — Yoked surface codes are concatenations of QMDPC codes with rotated surface codes.



  • Hypergraph product (HGP) code — Rotated code can be obtained from hypergraph product of two cyclic binary cyclic codes with palindromic generator polynomial ([3], Ex. 7).
  • Heavy-hexagon code — A rotated surface code can be mapped onto a heavy square lattice, resulting in a code similar to the heavy-hexagon code [12].
  • \([[4,2,2]]\) CSS code — The subcodes \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\) [13], \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [14], \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [15], and \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) [16] of the \([[4,2,2]]\) code are small planar rotated surface codes.
  • Ball-Verstraete-Cirac (BVC) code — An appropriately chosen stabilizer generator set for the BVC code contains the stabilizers of the rotated surface code [17].
  • 3D surface code — There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [18].
  • XZZX surface code — The XZZX code is obtained from the rotated surface code by applying Hadamard gates on a subset of qubits such that \(XXXX\) and \(ZZZZ\) generators are both mapped to \(XZXZ\). Both rotated and XZZX codes offer improved performance over the original surface code for biased noise [19].
  • Compass code — The surface-density compass code family interpolates between Bacon-Shor codes and rotated surface codes.
  • Subsystem rotated surface code


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Nikolas P. Breuckmann, private communication, 2022
Anthony Leverrier, Mapping the toric code to the rotated toric code, 2022.
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C. K. Andersen et al., “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020) arXiv:1912.09410 DOI
“Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI
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E. Huang et al., “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 DOI
D. Forlivesi, L. Valentini, and M. Chiani, “Logical Error Rates of XZZX and Rotated Quantum Surface Codes”, (2023) arXiv:2312.17057
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Zoo Code ID: rotated_surface

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“Rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_rotated_surface, title={Rotated surface code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.