# XY surface code[1]

## Description

Also called the tailored surface code (TSC). Non-CSS derivative of the surface code whose generators are \(XXXX\) and \(YYYY\), obtained by mapping \(Z \to Y\) in the surface code.

## Protection

As a stabilizer code, \([[n=O(d^2), k=O(1), d]]\).

## Code Capacity Threshold

\(50\%\) at infinite \(Z\) bias with maximum-likelihood decoder [2].\(18.7\%\) for standard depolarising noise with maximum-likelihood decoder [2].

## Threshold

\(6.32(3)\%\) for infinite \(Z\) bias, and thresholds of \(\sim 5\%\) for \(Z\) bias around \(\eta = 100\) using a variant of the minimum-weight perfect matching decoder [3].

## Parent

- Clifford-deformed surface code (CDSC) — XY code is obtained from the surface code by applying \(H\sqrt{Z}H\) to all qubits, thereby exchaning \(Z\leftrightarrow Y\).

## Cousin

- Heavy-hexagon code — XY surface code can be adapted for a heavy-hexagonal lattice [4].

## References

- [1]
- D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, “Ultrahigh Error Threshold for Surface Codes with Biased Noise”, Physical Review Letters 120, (2018) arXiv:1708.08474 DOI
- [2]
- D. K. Tuckett et al., “Tailoring Surface Codes for Highly Biased Noise”, Physical Review X 9, (2019) arXiv:1812.08186 DOI
- [3]
- D. K. Tuckett et al., “Fault-Tolerant Thresholds for the Surface Code in Excess of 5% Under Biased Noise”, Physical Review Letters 124, (2020) arXiv:1907.02554 DOI
- [4]
- Y. Kim, J. Kang, and Y. Kwon, “Design of Quantum error correcting code for biased error on heavy-hexagon structure”, (2022) arXiv:2211.14038

## Page edit log

- Victor V. Albert (2022-01-20) — most recent
- Arpit Dua (2022-01-19)

## Cite as:

“XY surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/xysurface