Rotated surface code[1][2][3][4]


Also called a checkerboard code. CSS 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.


The \([[L^2,1,L]]\) variant of this family 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.


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 be fault-tolerant to syndrome qubit errors [6][7][4].


  • Kitaev surface code — Rotated surface codes can be obtained using the same procedure as for the original surface codes but considering slightly different combinatorial surfaces [1][8] than those considered in the original proposal.
  • Quantum Tanner code — Specializing the quantum Tanner construction to the surface code yields the rotated surface code [9][10].
  • Hierarchical code — Hierarchical code is a concatenation of a constant-rate QLDPC code with a rotated surface code.




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J. T. Anderson, “Homological Stabilizer Codes”, (2011) arXiv:1107.3502
A. A. Kovalev and L. P. Pryadko, “Improved quantum hypergraph-product LDPC codes”, 2012 IEEE International Symposium on Information Theory Proceedings (2012) arXiv:1202.0928 DOI
Y. Tomita and K. M. Svore, “Low-distance surface codes under realistic quantum noise”, Physical Review A 90, (2014) arXiv:1404.3747 DOI
C. Chamberland et al., “Techniques for combining fast local decoders with global decoders under circuit-level noise”, (2022) arXiv:2208.01178
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
N. Delfosse, P. Iyer, and D. Poulin, “Generalized surface codes and packing of logical qubits”, (2016) arXiv:1606.07116
Nikolas P. Breuckmann, private communication, 2022
Anthony Leverrier, Mapping the toric code to the rotated toric code, 2022.
C. Chamberland et al., “Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits”, Physical Review X 10, (2020) arXiv:1907.09528 DOI
A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 DOI
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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Zoo Code ID: rotated_surface

Cite as:
“Rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_rotated_surface, title={Rotated surface code}, booktitle={The Error Correction Zoo}, year={2023}, 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.), 2023.