Also known as Wen plaquette model.

## Description

Non-CSS variant of the rotated surface code whose generators are \(XZZX\) 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).

XZZX toric code often either refers to the construction on the two-dimensional torus or is an alternative name for the general construction. Twisted XZZX toric code refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions. The construction on surfaces with boundaries is often called the XZZX planar code.

Stabilizer generators for this code are shown in Figure I.

## Protection

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

## Decoding

MWPM 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. [5]. Decoding complexity scales as order \(O(n^3)\) because the code is non-CSS [6][5; Supplement].

## Code Capacity Threshold

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.\(50\%\) threshold for noise infinitely biased towards \(X\) or \(Y\) or \(Z\) errors using a maximum-likelihood decoder.Depolarizing noise: \(18.7(1)\%\) under tensor-network decoder [7] and \(17.5\%\) under AMBP4 [8].

## Threshold

\(\approx 4.5\%\) using minimum-weight perfect matching decoder for depolarizing noise (bias \(\eta=0.5\)); \(\approx 10\%\) for infinite \(Z\) bias.\(4.15\%\) when \(98\%\) of depolarizing errors are converted into erasure errors with union-find decoder on a planar code, vs. \(0.937\%\) for pure depolarizing noise. In Rydberg atomic devices, the dominant source of noise is spontaneous decay into detectable energy levels outside of the computational subspace. Since that decay occurs in a Rydberg level that is accessible from only of the hyperfine states used for storage, the resulting channel is biased erasure [9].\(0.817\%\) and \(0.940\%\) with minimum-weight perfect matching and belief-matching decoder, respectively, for biased circuit-level noise [10].

## Realizations

Superconducting circuits: Distance-five 25-qubit code implemented on a superconducting quantum processor by Google Quantum AI [11]. 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 [12]. Leakage errors have been handled in a separate work in a distance-three code [13]. Google Quantum AI follow-up experiment realizing distance-5 and distance-7 codes with 100 rounds of correction using the Libra and transformer-based decoders. The logical error rate is suppressed by a factor of \(\approx 2\), demonstrating beyond-break-even error correction with a block quantum code [14]. Rydberg atom arrays: Lukin group [15]. Transversal CNOT gates performed on distance \(3\), \(5\), and \(7\) codes.

## 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].

## Parents

- Twist-defect surface code — XZZX toric and planar codes can be treated in the general twist-defect surface code formalism [16].
- Abelian quantum-double stabilizer code
- Clifford-deformed surface code (CDSC) — The XZZX surface 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

- Twisted XZZX toric code — Imposing twisted (a.k.a. shifted) boundary conditions on the toric XZZX code yields the twisted XZZX code [2; Ex. 11 and Fig. 3][16; Fig. 6].

## Cousins

- Rotated surface code — The 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 \(XZXZ\). Both rotated and XZZX codes offer improved performance over the original surface code for biased noise [17].
- Chamon model code — The Chamon model code can be obtained from a particular hypergraph product of three repetition codes [18]; see [19; Sec. 3.4]. Using only two repetition codes yields the XZZX code, making that code a 2D analogue of the Chamon code [19; Sec. 2].
- Fracton stabilizer code — Subsystem symmetries play a role in finite-bias decoders for both XZZX and fracton codes [20].
- Abelian quantum-double stabilizer code — The XZZX surface code is an example of \(\mathbb{Z}_2\) topological order as manifest in the Wen plaquette model [1].
- Heavy-hexagon code — XZZX surface code can be adapted for a heavy-hexagonal lattice [21].
- Cluster-state code — XZZX surface code can be foliated for a noise-bias preserving MBQC [22] or FBQC [23] protocol; see also [24].
- GKP-surface code — GKP codes have been concatenated with XZZX surface codes [25].
- Asymmetric quantum code — The XZZX surface code can be foliated for a noise-bias preserving MBQC [22] or FBQC [23] protocol; see also [24].
- Derby-Klassen (DK) code — The DK code encodes fermions into excitations of the Wen plaquette model [26].
- XYZ color code — The XZZX surface (XYZ color) is a non-CSS analogue of the rotated surface (6.6.6 color) code such that the two codes are related by single-qubit Clifford rotations.

## References

- [1]
- X.-G. Wen, “Quantum Orders in an Exact Soluble Model”, Physical Review Letters 90, (2003) arXiv:quant-ph/0205004 DOI
- [2]
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- J. P. Bonilla Ataides et al., “The XZZX surface code”, Nature Communications 12, (2021) arXiv:2009.07851 DOI
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- 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
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- K.-Y. Kuo and C.-Y. Lai, “Comparison of 2D topological codes and their decoding performances”, 2022 IEEE International Symposium on Information Theory (ISIT) (2022) arXiv:2202.06612 DOI
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- 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
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- K.-Y. Kuo and C.-Y. Lai, “Exploiting degeneracy in belief propagation decoding of quantum codes”, npj Quantum Information 8, (2022) arXiv:2104.13659 DOI
- [9]
- 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
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- O. Higgott et al., “Improved decoding of circuit noise and fragile boundaries of tailored surface codes”, (2023) arXiv:2203.04948
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- R. Acharya et al., “Suppressing quantum errors by scaling a surface code logical qubit”, (2022) arXiv:2207.06431
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- T. I. Andersen et al., “Non-Abelian braiding of graph vertices in a superconducting processor”, (2023) arXiv:2210.10255
- [13]
- K. C. Miao et al., “Overcoming leakage in quantum error correction”, Nature Physics 19, 1780 (2023) arXiv:2211.04728 DOI
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- R. Acharya et al., “Quantum error correction below the surface code threshold”, (2024) arXiv:2408.13687
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- D. Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays”, Nature (2023) arXiv:2312.03982 DOI
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- R. Sarkar and T. J. Yoder, “A graph-based formalism for surface codes and twists”, Quantum 8, 1416 (2024) arXiv:2101.09349 DOI
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- D. Forlivesi, L. Valentini, and M. Chiani, “Logical Error Rates of XZZX and Rotated Quantum Surface Codes”, (2023) arXiv:2312.17057
- [18]
- Maurice, Denise. Codes correcteurs quantiques pouvant se décoder itérativement. Diss. Université Pierre et Marie Curie-Paris VI, 2014.
- [19]
- A. Leverrier, S. Apers, and C. Vuillot, “Quantum XYZ Product Codes”, Quantum 6, 766 (2022) arXiv:2011.09746 DOI
- [20]
- B. J. Brown and D. J. Williamson, “Parallelized quantum error correction with fracton topological codes”, Physical Review Research 2, (2020) arXiv:1901.08061 DOI
- [21]
- Y. Kim, J. Kang, and Y. Kwon, “Design of Quantum error correcting code for biased error on heavy-hexagon structure”, (2022) arXiv:2211.14038
- [22]
- 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
- [23]
- H. Bombín et al., “Increasing error tolerance in quantum computers with dynamic bias arrangement”, (2023) arXiv:2303.16122
- [24]
- A. M. Stephens, W. J. Munro, and K. Nemoto, “High-threshold topological quantum error correction against biased noise”, Physical Review A 88, (2013) arXiv:1308.4776 DOI
- [25]
- J. Zhang, Y.-C. Wu, and G.-P. Guo, “Concatenation of the Gottesman-Kitaev-Preskill code with the XZZX surface code”, Physical Review A 107, (2023) arXiv:2207.04383 DOI
- [26]
- R. W. Chien and J. D. Whitfield, “Custom fermionic codes for quantum simulation”, (2020) arXiv:2009.11860

## Page edit log

- Eric Huang (2024-03-18) — most recent
- Victor V. Albert (2022-07-15)
- 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.), 2024. https://errorcorrectionzoo.org/c/xzzx