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]
- 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
- [3]
- B. M. Terhal, F. Hassler, and D. P. DiVincenzo, “From Majorana fermions to topological order”, Physical Review Letters 108, (2012) arXiv:1201.3757 DOI
- [4]
- J. P. Bonilla Ataides et al., “The XZZX surface code”, Nature Communications 12, (2021) arXiv:2009.07851 DOI
- [5]
- 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
- [6]
- 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
- [7]
- 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
- [8]
- 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
- [10]
- O. Higgott et al., “Improved decoding of circuit noise and fragile boundaries of tailored surface codes”, (2023) arXiv:2203.04948
- [11]
- R. Acharya et al., “Suppressing quantum errors by scaling a surface code logical qubit”, (2022) arXiv:2207.06431
- [12]
- 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
- [14]
- R. Acharya et al., “Quantum error correction below the surface code threshold”, (2024) arXiv:2408.13687
- [15]
- D. Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays”, Nature (2023) arXiv:2312.03982 DOI
- [16]
- R. Sarkar and T. J. Yoder, “A graph-based formalism for surface codes and twists”, Quantum 8, 1416 (2024) arXiv:2101.09349 DOI
- [17]
- 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