XZZX surface code[14] 

Also known as Wen plaquette model.


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.

Figure I: Stabilizer generators of a XZZX planar code with open boundaries. The generators are four-body \(XZZX\) operators on the corners of squares in the bulk and two-body \(XZ\) operators on the boundaries.


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


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].


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


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].Rydberg atom arrays: Lukin group [14]. Transversal CNOT gates performed on distance \(3\), \(5\), and \(7\) codes.


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].




  • 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 [16].
  • Chamon model code — The Chamon model code can be obtained from a particular hypergraph product of three repetition codes [17]; see [18; Sec. 3.4]. Using only two repetition codes yields the XZZX code, making that code a 2D analogue of the Chamon code [18; Sec. 2].
  • Fracton stabilizer code — Subsystem symmetries play a role in finite-bias decoders for both XZZX and fracton codes [19].
  • 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 [20].
  • Cluster-state code — XZZX surface code can be foliated for a noise-bias preserving MBQC [21] or FBQC [22] protocol; see also [23].
  • Asymmetric quantum code — The XZZX surface code can be foliated for a noise-bias preserving MBQC [21] or FBQC [22] protocol; see also [23].
  • Derby-Klassen (DK) code — The DK code encodes fermions into excitations of the Wen plaquette model [24].
  • 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.


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J. P. Bonilla Ataides et al., “The XZZX surface code”, Nature Communications 12, (2021) arXiv:2009.07851 DOI
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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Maurice, Denise. Codes correcteurs quantiques pouvant se décoder itérativement. Diss. Université Pierre et Marie Curie-Paris VI, 2014.
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