XYZ\(^2\) hexagonal stabilizer code

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 honeycomb tiling 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].Sequential decoder [4].

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.\(\approx 18\%\) for depolarizing noise under maximum-likelihood decoding.\(18.3\%\) under biased noise [4].

Notes

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

Cousins

XZZX surface code— The XYZ\(^2\) hexagonal stabilizer code can be viewed as a concatenation of the \(YZZY\) surface code with the \([[2,1]]\) phase-flip repetition code [4].
Quantum repetition code— The XYZ\(^2\) hexagonal stabilizer code can be viewed as a concatenation of the \(YZZY\) surface code with the \([[2,1]]\) phase-flip repetition code [4].
Asymmetric quantum code— The XYZ\(^2\) hexagonal stabilizer code has high thresholds under biased noise [4].

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

Matching codeLattice stabilizer QLDPC Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Parents

Matching code

Concatenated qubit codeQubit Concatenated quantum QECC Quantum

The XYZ\(^2\) hexagonal stabilizer code can be viewed as a concatenation of the \(YZZY\) surface code with the \([[2,1]]\) phase-flip repetition code [4].

XYZ\(^2\) hexagonal stabilizer code

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, A. Orekhov, P. W. Hybelius, A. K. Wisakanto, B. Srivastava, A. F. Kockum, and M. Granath, “Error-rate-agnostic decoding of topological stabilizer codes”, Physical Review A 105, (2022) arXiv:2112.01977 DOI
[4]: B. Srivastava, Y. Xiao, A. F. Kockum, B. Criger, and M. Granath, “Sequential decoding of the XYZ\(^2\) hexagonal stabilizer code”, (2025) arXiv:2505.03691

Page edit log

Basudha Srivastava (2022-03-16) — most recent
Victor V. Albert (2022-03-16)

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/edit/main/codes/quantum/qubits/stabilizer/topological/matching/xyz_hexagonal.yml.