# XZZX surface code[1]

## Description

Also called a rotated surface code. Non-CSS derivative of the surface code whose generators are \(XZXZ\) Pauli strings associated, clock-wise, to the vertices of each face of a two-dimensional lattice (with a qubit located at each vertex of the tessellation).

## Protection

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

## Decoding

Minimum-weight perfect matching decoder, which can be used for \(X\) and \(Z\) noise. For \(Y\) noise, a variant of the matching decoder could be used like it is used for the XY code in Ref. [2].

## Code Capacity Threshold

\(50\%\) threshold for noise infinitely biased towards \(X\) or \(Y\) or \(Z\) errors using a maximum-likelihood decoder.For large but finite \(X\)- or \(Z\)-biased noise, the code's thresholds exceed the zero-rate hashing bound. The difference of the threshold from the hashing bound exceeds \(2.9\%\) at a \(Z\) or \(X\) bias of 300.\(18.7\%\) for standard depolarising noise with maximum-likelihood decoder.\(0.817\%\) and \(0.940\%\) with minimum-weight perfect matching and belief-matching decoder, respectively, for biased circuit-level noise [3].\(4.15\%\) when \(98\%\) of depolarizing errors are coverted into erasure errors with union-find decoder on a planar code, vs. \(0.937\%\) for pure depolarizing noise. In Rydberg atomic devices, erasure conversion during gates is promising because the dominant source of noise is spontaneous decay into detectable energy levels outside of the computational subspace [4].

## Threshold

\(\sim 4.5\%\) using minimum-weight perfect matching decoder for depolarizing noise (bias \(\eta=0.5\)); \(\sim 10\%\) for infinite \(Z\) bias.

## Notes

A single \(X\) or \(Z\) error gives rise to two nearby defects, which can be viewed as endpoints of a string. That way, multiple \(Z\) errors can be decomposed into a combination of diagonal strings.Originally formulated as an example of \(\mathbb{Z}_2\) topological order in the Wen plaquette model [5].

## Parent

- Clifford-deformed surface code (CDSC) — XZZX code is obtained from the surface code by applying Hadamard gates on a subset of qubits such that \(XXXX\) and \(ZZZZ\) generators are both mapped to \(XZXZ\).

## Cousins

- Fracton code — Subsystem symmetries play a role in finite-bias decoders for both codes [6].
- Abelian topological code — Example of \(\mathbb{Z}_2\) topological order in the Wen plaquette model [5].

## Zoo code information

## References

- [1]
- J. P. Bonilla Ataides et al., “The XZZX surface code”, Nature Communications 12, (2021). DOI; 2009.07851
- [2]
- D. K. Tuckett et al., “Fault-Tolerant Thresholds for the Surface Code in Excess of 5% Under Biased Noise”, Physical Review Letters 124, (2020). DOI; 1907.02554
- [3]
- Oscar Higgott et al., “Fragile boundaries of tailored surface codes”. 2203.04948
- [4]
- Yue Wu et al., “Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays”. 2201.03540
- [5]
- X.-G. Wen, “Quantum Orders in an Exact Soluble Model”, Physical Review Letters 90, (2003). DOI; quant-ph/0205004
- [6]
- B. J. Brown and D. J. Williamson, “Parallelized quantum error correction with fracton topological codes”, Physical Review Research 2, (2020). DOI; 1901.08061

## Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/surface/xzzx.yml.