[Jump to code hierarchy]

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 Fig. 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 correspond 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 \([[L^2,2,L]]\) periodic variant is a rotated toric or checkerboard code, whose smallest example is the \([[4,2,2]]\) code. Non-bipartite rotated toric codes with odd distance include the family \([[t^2+(t+1)^2,1,2t+1]]\) [3; Exam. 4].

Encoding

Unitary encoder based on code conversion between rotated and regular surface codes [5].

Transversal Gates

Fold-transversal \(S\) gate [6,7].

Gates

Injection of the \(|Y\rangle\) state [8].

Decoding

Only certain syndrome extraction schedules are distance-preserving [4].Local neural-network using 3D convolutions, combined with a separate global decoder [9].Iterative CNOT decoder [10].Fault-tolerant BP (FTBP) decoder [11].

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 [4,12,13].

Threshold

Thresholds for various amounts of erasure, Pauli, correlated, and measurement noise are known [14,15].

Realizations

Teleportation transition of distance-seven rotated surface-code states on a 125-qubit superconducting processor [16].

Cousins

  • Hypergraph product (HGP) code— Periodic checkerboard or rotated-toric codes on the same lattice can be obtained from hypergraph products of two cyclic linear binary codes with palindromic check polynomials [3; Sec. IV.D].
  • Cyclic linear binary code— Periodic checkerboard or rotated-toric codes on the same lattice can be obtained from hypergraph products of two cyclic linear binary codes with palindromic check polynomials [3; Sec. IV.D].
  • 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 [17].
  • 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.
  • Plaquette Ising code— The 2D 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 [18].
  • GKP-surface code— GKP codes have been concatenated with rotated surface codes [19–22].
  • Tensor-network code— A tensor-network based modification of the rotated surface code improves performance against depolarizing noise by \(\approx 2\%\) [23].
  • Ball-Verstraete-Cirac (BVC) code— An appropriately chosen stabilizer generator set for the BVC code contains the stabilizers of the rotated surface code [24].
  • Toric code— Rotating the square lattice by \(\pi/4\) and choosing periodicity vectors on the rotated checkerboard lattice yields periodic checkerboard or rotated-toric variants with the same \([[L^2,2,L]]\) scaling, as well as non-bipartite odd-distance families with parameters \([[t^2+(t+1)^2,1,2t+1]]\) [3; Sec. III].
  • 3D surface code— There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [25].
  • 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 [26].
  • Compass code— The surface-density compass code family interpolates between Bacon-Shor codes and rotated surface codes.
  • Subsystem rotated surface code— Subsystem rotated surface codes are subsystem versions of rotated surface codes.

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 Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Homological code
Kitaev surface codeCDSC Twist-defect surface QLDPC Qubit CSS Generalized homological-product Lattice stabilizer Stabilizer 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 [1,27][28; Fig. 8]. Applying the quantum Tanner transformation to the surface code yields the rotated surface code [29,30]. The rotated surface code presents certain savings over the original surface code [31].
Quantum Tanner codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Applying the quantum Tanner transformation to the surface code yields the rotated surface code [29,30].
Rotated surface code
Children
\([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code
The \([[4,1,2]]\) LNCY code is a small planar rotated surface code [32–35].
\([[4,2,2]]\) Four-qubit code
The \([[4,2,2]]\) code is the smallest rotated toric code [36].
\([[5,1,2]]\) rotated surface code
\([[9,1,3]]\) Surface-17 code

References

[1]
H. Bombin and M. A. Martin-Delgado, “Optimal resources for topological two-dimensional stabilizer codes: Comparative study”, Physical Review A 76, (2007) arXiv:quant-ph/0703272 DOI
[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]
P.-K. Tsai and S. Puri, “A Unitary Encoder for Surface Codes”, (2025) arXiv:2506.04084
[6]
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
[7]
K. H. Wan and Z. Zhong, “Pauli webs spun by transversal \(|Y\rangle\) state initialisation”, (2025) arXiv:2502.00957
[8]
K. H. Wan and Z. Zhong, “Pauli web of the \(|Y\rangle\) state surface code injection”, (2025) arXiv:2501.15566
[9]
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
[10]
K. H. Wan, M. Webber, A. G. Fowler, and W. K. Hensinger, “An iterative transversal CNOT decoder”, (2025) arXiv:2407.20976
[11]
K.-Y. Kuo and C.-Y. Lai, “Fault-Tolerant Belief Propagation for Practical Quantum Memory”, (2024) arXiv:2409.18689
[12]
E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[13]
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
[14]
S. Gu, Y. Vaknin, A. Retzker, and A. Kubica, “Optimizing Quantum Error-Correction Protocols with Erasure Qubits”, PRX Quantum 6, (2025) arXiv:2408.00829 DOI
[15]
J. F Kam, S. Gicev, K. Modi, A. Southwell, and M. Usman, “Detrimental non-Markovian errors for surface code memory”, Quantum Science and Technology 10, 035060 (2025) arXiv:2410.23779 DOI
[16]
Y. Zou et al., “Teleportation transition of surface codes on a superconducting quantum processor”, (2026) arXiv:2602.21293
[17]
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
[18]
C. Chamberland et al., “Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes”, PRX Quantum 3, (2022) arXiv:2012.04108 DOI
[19]
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
[20]
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
[21]
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
[22]
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
[23]
T. Farrelly, D. K. Tuckett, and T. M. Stace, “Local tensor-network codes”, New Journal of Physics 24, 043015 (2022) arXiv:2109.11996 DOI
[24]
C. Derby. Compact fermion to qubit mappings for quantum simulation. PhD thesis, UCL (University College London), 2023
[25]
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
[26]
D. Forlivesi, L. Valentini, and M. Chiani, “Logical Error Rates of XZZX and Rotated Quantum Surface Codes”, (2023) arXiv:2312.17057
[27]
N. Delfosse, P. Iyer, and D. Poulin, “Generalized surface codes and packing of logical qubits”, (2016) arXiv:1606.07116
[28]
R. Sarkar and T. J. Yoder, “A graph-based formalism for surface codes and twists”, Quantum 8, 1416 (2024) arXiv:2101.09349 DOI
[29]
N. P. Breuckmann, private communication, 2022
[30]
A. Leverrier, Mapping the toric code to the rotated toric code, 2022
[31]
A. R. O’Rourke and S. Devitt, “Compare the Pair: Rotated vs. Unrotated Surface Codes at Equal Logical Error Rates”, (2024) arXiv:2409.14765
[32]
A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 DOI
[33]
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
[34]
“Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
[35]
J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI
[36]
A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

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} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/rotated_surface

Cite as:

“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/rotated_surface/rotated_surface/rotated_surface.yml.