## Description

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.

## Protection

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.

## Decoding

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].

## Parents

- 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.

## Child

## Cousins

- Hypergraph product 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 [11].
- \([[4,2,2]]\) CSS code — Various \([[4,1,2]]\) subcodes are small rotated planar codes [12][13][14][15].
- Subsystem rotated surface code
- XZZX surface code — 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 \(XZXZ\).

## 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”, (2022) arXiv:2208.01178
- [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]
- Nikolas P. Breuckmann, private communication, 2022
- [10]
- Anthony Leverrier, Mapping the toric code to the rotated toric code, 2022.
- [11]
- C. Chamberland et al., “Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits”, Physical Review X 10, (2020) arXiv:1907.09528 DOI
- [12]
- A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 DOI
- [13]
- C. K. Andersen et al., “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020) arXiv:1912.09410 DOI
- [14]
- “Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
- [15]
- J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI

## Page edit log

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

## Cite as:

“Rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/rotated_surface