XYZ\(^2\) hexagonal stabilizer code[1,2] 

Description

An instance of the matching code based on the Kitaev honeycomb model. It is described on a hexagonal lattice with \(XYZXYZ\) stabilizers on each hexagonal plaquette. Each vertical pair of qubits has an \(XX\), \(YY\), or \(ZZ\) link stabilizer depending on the orientation of the plaquette stabilizers.

Protection

As a stabilizer code with boundaries, protects a single qubit with parameters \([[2 d^2, 1, d]]\).

Decoding

Maximum-likelihood decoding using the EWD decoder [3].

Code Capacity Threshold

\(50\%\) for pure \(Z\), \(Y\), or \(Z\) noise under maximum-likelihood decoding.Threshold matches that of the \(XZZX\) code for various bias levels of \(X\), \(Y\), or \(Z\) biased noise under maximum-likelihood decoding.\(\sim 18\%\) for depolarizing noise under maximum-likelihood decoding.

Notes

Isolated \(X\), \(Y\), and \(Z\) errors lead to unidirectional pairs of plaquette defects along the three directions of the triangular lattice.

Parent

References

[1]
J. R. Wootton, “Hexagonal matching codes with two-body measurements”, Journal of Physics A: Mathematical and Theoretical 55, 295302 (2022) arXiv:2109.13308 DOI
[2]
B. Srivastava, A. Frisk Kockum, and M. Granath, “The XYZ2 hexagonal stabilizer code”, Quantum 6, 698 (2022) arXiv:2112.06036 DOI
[3]
K. Hammar et al., “Error-rate-agnostic decoding of topological stabilizer codes”, Physical Review A 105, (2022) arXiv:2112.01977 DOI
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Zoo Code ID: xyz_hexagonal

Cite as:
“XYZ\(^2\) hexagonal stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/xyz_hexagonal
BibTeX:
@incollection{eczoo_xyz_hexagonal,
  title={XYZ\(^2\) hexagonal stabilizer code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/xyz_hexagonal}
}
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Cite as:

“XYZ\(^2\) hexagonal stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/xyz_hexagonal

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/stabilizer/topological/xyz_hexagonal.yml.