## Description

Non-CSS variant of the rotated 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. [4].

## 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 [5].\(4.15\%\) 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 [6].

## Threshold

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

## Realizations

Distance-five 25-qubit code implemented on a superconducting quantum processor by Google Quantum AI [7]. This code outperformed the average of several instances of the smaller distance-three 9-qubit \(XZZX\) variant of the surface-17 code realized on the same device, both in terms of logical error probability over 25 cycles and in terms of logical error per cycle. This increase in error-correcting capabilities while using more physical qubits supports the notion of an error threshold. Braiding of defects has been demonstrated for the distance-five code [8]. Leakage errors have been handled in a separate work in a distance-three code [9].

## 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 [1].

## Parent

- Clifford-deformed surface code (CDSC) — 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 \(XZXZ\).

## Child

- Five-qubit perfect code — The five-qubit code is the smallest XZZX surface code [3; Ex. 11][10; Ex. 3].

## Cousins

- Rotated 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 \(XZXZ\).
- Fracton code — Subsystem symmetries play a role in finite-bias decoders for both codes [11].
- Abelian topological code — Example of \(\mathbb{Z}_2\) topological order in the Wen plaquette model [1].
- Heavy-hexagon code — XZZX surface code can be adapted for a heavy-hexagonal lattice [12].
- Cluster-state code — XZZX surface code can be foliated for a noise-bias preserving MBQC protocol [13].
- Cluster-state code — XZZX surface code can be foliated for a noise-bias preserving MBQC protocol [14].

## References

- [1]
- X.-G. Wen, “Quantum Orders in an Exact Soluble Model”, Physical Review Letters 90, (2003) arXiv:quant-ph/0205004 DOI
- [2]
- J. P. Bonilla Ataides et al., “The XZZX surface code”, Nature Communications 12, (2021) arXiv:2009.07851 DOI
- [3]
- A. A. Kovalev, I. Dumer, and L. P. Pryadko, “Design of additive quantum codes via the code-word-stabilized framework”, Physical Review A 84, (2011) arXiv:1108.5490 DOI
- [4]
- 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
- [5]
- O. Higgott et al., “Fragile boundaries of tailored surface codes and improved decoding of circuit-level noise”, (2022) arXiv:2203.04948
- [6]
- Y. Wu et al., “Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays”, Nature Communications 13, (2022) arXiv:2201.03540 DOI
- [7]
- R. Acharya et al., “Suppressing quantum errors by scaling a surface code logical qubit”, (2022) arXiv:2207.06431
- [8]
- T. I. Andersen et al., “Observation of non-Abelian exchange statistics on a superconducting processor”, (2022) arXiv:2210.10255
- [9]
- K. C. Miao et al., “Overcoming leakage in scalable quantum error correction”, (2022) arXiv:2211.04728
- [10]
- A. A. Kovalev and L. P. Pryadko, “Quantum Kronecker sum-product low-density parity-check codes with finite rate”, Physical Review A 88, (2013) arXiv:1212.6703 DOI
- [11]
- B. J. Brown and D. J. Williamson, “Parallelized quantum error correction with fracton topological codes”, Physical Review Research 2, (2020) arXiv:1901.08061 DOI
- [12]
- Y. Kim, J. Kang, and Y. Kwon, “Design of Quantum error correcting code for biased error on heavy-hexagon structure”, (2022) arXiv:2211.14038
- [13]
- J. Claes, J. E. Bourassa, and S. Puri, “Tailored cluster states with high threshold under biased noise”, npj Quantum Information 9, (2023) arXiv:2201.10566 DOI
- [14]
- K. Sahay, J. Claes, and S. Puri, “Tailoring fusion-based error correction for high thresholds to biased fusion failures”, (2022) arXiv:2301.00019

## Page edit log

- Victor V. Albert (2022-07-15) — most recent
- Victor V. Albert (2022-03-24)
- Arpit Dua (2022-01-19)
- Marianna Podzorova (2021-12-13)

## Cite as:

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