Rotated surface code[14] 

Also known as 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 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.

Protection

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.

Decoding

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].

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 [1,8][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.

Child

Cousins

  • 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

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 (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]
C. Chamberland et al., “Techniques for combining fast local decoders with global decoders under circuit-level noise”, Quantum Science and Technology 8, 045011 (2023) arXiv:2208.01178 DOI
[6]
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[7]
A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
[8]
N. Delfosse, P. Iyer, and D. Poulin, “Generalized surface codes and packing of logical qubits”, (2016) arXiv:1606.07116
[9]
R. Sarkar and T. J. Yoder, “A graph-based formalism for surface codes and twists”, (2023) arXiv:2101.09349
[10]
Nikolas P. Breuckmann, private communication, 2022
[11]
Anthony Leverrier, Mapping the toric code to the rotated toric code, 2022.
[12]
C. Chamberland et al., “Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits”, Physical Review X 10, (2020) arXiv:1907.09528 DOI
[13]
A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 DOI
[14]
C. K. Andersen et al., “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020) arXiv:1912.09410 DOI
[15]
“Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
[16]
J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI
[17]
Derby, Charles. Compact fermion to qubit mappings for quantum simulation. Diss. UCL (University College London), 2023.
[18]
E. Huang et al., “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 DOI
[19]
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

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.