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Rotated surface code[1–4]

Alternative names: Checkerboard code, Medial surface code, Rectified surface code.

Description

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 weight-four (four-body) operators on the corners of squares in the bulk and weight-two (two-body) operators on the boundaries. Red regions correspond to \(X\) operators while blue regions to \(Z\) operators.

Protection

The \([[L^2,1,L]]\) planar rotated surface code 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. The \([[L^2,2,L]]\) rotated toric code includes the \([[4,2,2]]\) code as its smallest example.

Transversal Gates

Fold-transversal \(S\) gate [5,6].

Gates

Injection of the \(|Y\rangle\) state [7].

Decoding

Only certain syndrome extraction schedules are distance-preserving [4].Local neural-network using 3D convolutions, combined with a separate global decoder [8].Iterative CNOT decoder [9].Fault-tolerant BP (FTBP) decoder [10].

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 [4,11,12].

Threshold

Thresholds for various amounts of erasure, Pauli, correlated, and measurement noise are known [13,14].

Cousins

  • Hypergraph product (HGP) code— The rotated code can be obtained from hypergraph product of two cyclic linear binary codes with palindromic generator polynomial [3; Exam. 7].
  • Cyclic linear binary code— The rotated code can be obtained from hypergraph product of two cyclic linear binary codes with palindromic generator polynomial [3; Exam. 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 [15].
  • Plaquette Ising code— The plaquette Ising model can be thought of as the rotated surface code whose \(X\)-type stabilizer generators have been converted to \(Z\)-type stabilizer generators.
  • Concatenated cat code— Cat codes have been concatenated with rotated surface codes [16].
  • GKP-surface code— GKP codes have been concatenated with rotated surface codes [17–21].
  • Tensor-network code— A tensor-network based modification of the rotated surface code improves performance against depolarizing noise by \(\approx 2\%\) [22].
  • Ball-Verstraete-Cirac (BVC) code— An appropriately chosen stabilizer generator set for the BVC code contains the stabilizers of the rotated surface code [23].
  • 3D surface code— There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [24].
  • 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 [25].
  • Compass code— The surface-density compass code family interpolates between Bacon-Shor codes and rotated surface codes.
  • Subsystem rotated surface code

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Homological code
Kitaev surface codeCDSC Twist-defect surface Lattice stabilizer Generalized homological-product QLDPC CSS Stabilizer Qubit Abelian topological Topological Hamiltonian-based QECC Quantum
Parents
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 [1,26][27; Fig. 8]. Applying the quantum Tanner transformation to the surface code yields the rotated surface code [28,29]. The rotated surface code presents certain savings over the original surface code [30].
Quantum Tanner codeQubit Generalized homological-product QLDPC CSS Stabilizer Hamiltonian-based Single-shot QECC Quantum
Applying the quantum Tanner transformation to the surface code yields the rotated surface code [28,29].
Hierarchical codeQLDPC Stabilizer Hamiltonian-based Qubit Concatenated quantum QECC Quantum
Hierarchical codes are concatenations of constant-rate QLDPC codes with rotated surface codes.
Yoked surface codeCSS QLDPC Stabilizer Qubit Concatenated quantum Hamiltonian-based QECC Quantum
Yoked surface codes are concatenations of QMDPC codes with rotated surface codes.
Rotated surface code
Children
\([[4,2,2]]\) Four-qubit code
The \([[4,2,2]]\) code is the smallest rotated toric code. The subcodes \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\) [31], \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [32], \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [33], and \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) [34] are small planar rotated surface codes.
\([[5,1,2]]\) rotated surface code
\([[9,1,3]]\) Surface-17 code

References

[1]
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[2]
J. T. Anderson, “Homological Stabilizer Codes”, (2011) arXiv:1107.3502
[3]
A. A. Kovalev and L. P. Pryadko, “Improved quantum hypergraph-product LDPC codes”, 2012 IEEE International Symposium on Information Theory Proceedings 348 (2012) arXiv:1202.0928 DOI
[4]
Y. Tomita and K. M. Svore, “Low-distance surface codes under realistic quantum noise”, Physical Review A 90, (2014) arXiv:1404.3747 DOI
[5]
Z.-H. Chen, M.-C. Chen, C.-Y. Lu, and J.-W. Pan, “Transversal Logical Clifford gates on rotated surface codes with reconfigurable neutral atom arrays”, (2024) arXiv:2412.01391
[6]
K. H. Wan and Z. Zhong, “Pauli webs spun by transversal \(|Y\rangle\) state initialisation”, (2025) arXiv:2502.00957
[7]
K. H. Wan and Z. Zhong, “Pauli web of the \(|Y\rangle\) state surface code injection”, (2025) arXiv:2501.15566
[8]
C. Chamberland, L. Goncalves, P. Sivarajah, E. Peterson, and S. Grimberg, “Techniques for combining fast local decoders with global decoders under circuit-level noise”, Quantum Science and Technology 8, 045011 (2023) arXiv:2208.01178 DOI
[9]
K. H. Wan, M. Webber, A. G. Fowler, and W. K. Hensinger, “An iterative transversal CNOT decoder”, (2025) arXiv:2407.20976
[10]
K.-Y. Kuo and C.-Y. Lai, “Fault-Tolerant Belief Propagation for Practical Quantum Memory”, (2024) arXiv:2409.18689
[11]
E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[12]
A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
[13]
S. Gu, Y. Vaknin, A. Retzker, and A. Kubica, “Optimizing quantum error correction protocols with erasure qubits”, (2024) arXiv:2408.00829
[14]
J. F. Kam, S. Gicev, K. Modi, A. Southwell, and M. Usman, “Detrimental non-Markovian errors for surface code memory”, (2024) arXiv:2410.23779
[15]
C. Chamberland, G. Zhu, T. J. Yoder, J. B. Hertzberg, and A. W. Cross, “Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits”, Physical Review X 10, (2020) arXiv:1907.09528 DOI
[16]
C. Chamberland et al., “Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes”, PRX Quantum 3, (2022) arXiv:2012.04108 DOI
[17]
K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
[18]
K. Noh and C. Chamberland, “Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code”, Physical Review A 101, (2020) arXiv:1908.03579 DOI
[19]
M. V. Larsen, C. Chamberland, K. Noh, J. S. Neergaard-Nielsen, and U. L. Andersen, “Fault-Tolerant Continuous-Variable Measurement-based Quantum Computation Architecture”, PRX Quantum 2, (2021) arXiv:2101.03014 DOI
[20]
K. Noh, C. Chamberland, and F. G. S. L. Brandão, “Low-Overhead Fault-Tolerant Quantum Error Correction with the Surface-GKP Code”, PRX Quantum 3, (2022) arXiv:2103.06994 DOI
[21]
M. Lin, C. Chamberland, and K. Noh, “Closest Lattice Point Decoding for Multimode Gottesman-Kitaev-Preskill Codes”, PRX Quantum 4, (2023) arXiv:2303.04702 DOI
[22]
T. Farrelly, D. K. Tuckett, and T. M. Stace, “Local tensor-network codes”, New Journal of Physics 24, 043015 (2022) arXiv:2109.11996 DOI
[23]
Derby, Charles. Compact fermion to qubit mappings for quantum simulation. Diss. UCL (University College London), 2023.
[24]
E. Huang, A. Pesah, C. T. Chubb, M. Vasmer, and A. Dua, “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 DOI
[25]
D. Forlivesi, L. Valentini, and M. Chiani, “Logical Error Rates of XZZX and Rotated Quantum Surface Codes”, (2023) arXiv:2312.17057
[26]
N. Delfosse, P. Iyer, and D. Poulin, “Generalized surface codes and packing of logical qubits”, (2016) arXiv:1606.07116
[27]
R. Sarkar and T. J. Yoder, “A graph-based formalism for surface codes and twists”, Quantum 8, 1416 (2024) arXiv:2101.09349 DOI
[28]
Nikolas P. Breuckmann, private communication, 2022
[29]
Anthony Leverrier, Mapping the toric code to the rotated toric code, 2022.
[30]
A. R. O’Rourke and S. Devitt, “Compare the Pair: Rotated vs. Unrotated Surface Codes at Equal Logical Error Rates”, (2024) arXiv:2409.14765
[31]
A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 DOI
[32]
C. K. Andersen, A. Remm, S. Lazar, S. Krinner, N. Lacroix, G. J. Norris, M. Gabureac, C. Eichler, and A. Wallraff, “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020) arXiv:1912.09410 DOI
[33]
“Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
[34]
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.), 2024. https://errorcorrectionzoo.org/c/rotated_surface
BibTeX:
@incollection{eczoo_rotated_surface, title={Rotated surface code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/rotated_surface} }
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“Rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/rotated_surface

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/rotated_surface/rotated_surface.yml.