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.

## 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

## Page edit log

- Eric Huang (2024-03-14) — most recent
- Marcus P da Silva (2023-03-20)
- Victor V. Albert (2022-07-30)

## Cite as:

“Rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/rotated_surface