Alternative names: 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. On a closed lattice, the Wen plaquette realization of the XZZX toric code has the same \(\mathbb{Z}_2\) topological order as the toric code, and translation by one lattice unit exchanges the \(e\) and \(m\) anyons [5; Appx. C].
Stabilizer generators for this code are shown in Fig. 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. [6]. Decoding complexity scales as order \(O(n^3)\) because the code is non-CSS [7][6; 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 [8] and \(17.5\%\) under AMBP4 [9].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. The dominant source of noise in neutral atom arrays 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 one of the hyperfine states used for storage, the resulting channel is biased erasure [10].\(0.817\%\) and \(0.940\%\) with minimum-weight perfect matching and belief-matching decoder, respectively, for biased circuit-level noise [11].Realizations
Superconducting circuits: Distance-five 25-qubit code implemented on a superconducting quantum processor by Google Quantum AI [12]. This code outperformed the average of several instances of the smaller distance-three nine-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 [13]. Leakage errors have been handled in a separate work in a distance-three code [14]. 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 [15]. Magic-state cultivation was demonstrated on a device by Google Quantum AI by code switching between a distance-three 6.6.6 color code and distance-five \(XZZX\) surface code and decoding with the Tesseract decoder [16]. Neutral atom arrays: Lukin group. Transversal CNOT gates performed on distance \(3\), \(5\), and \(7\) codes [17]. Below-threshold performance on distance \(3\) and \(5\) codes with multiple rounds of syndrome extraction and error correction [18].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].Popular summary of the Google Quantum AI above-threshold result in Quanta Magazine.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 [19].
- Chamon model code— The Chamon model code can be obtained from an XYZ product of three repetition codes [20]; see [21; Sec. 3.4]. Using only two repetition codes in the analogous 2D construction yields the XZZX code, making it a 2D analogue of the Chamon code [21; Sec. 2].
- Repetition code— The Chamon model code can be obtained from an XYZ product of three repetition codes [20]; see [21; Sec. 3.4]. Using only two repetition codes in the analogous 2D construction yields the XZZX code, making it a 2D analogue of the Chamon code [21; Sec. 2].
- Fracton stabilizer code— Subsystem symmetries play a role in finite-bias decoders for both XZZX and fracton codes [22]. The XZZX surface code resembles a Type-I fracton code with lineons in the limit of infinite noise bias [23].
- Heavy-hexagon code— XZZX surface code can be adapted for a heavy-hexagonal point set [24].
- Cluster-state code— XZZX surface code can be foliated for a noise-bias preserving MBQC [25] or FBQC [26] protocol; see also [27].
- Concatenated cat code— The four-component cat code can be concatenated with the XZZX code to yield a fusion-based computation scheme on a 2D lattice [28].
- GKP-surface code— GKP codes have been concatenated with XZZX surface codes [29].
- Asymmetric quantum code (AQC)— The XZZX surface code can be foliated for a noise-bias preserving MBQC [25] or FBQC [26] protocol; see also [27].
- Derby-Klassen (DK) code— The DK code encodes fermions into excitations of the Wen plaquette model [30].
- 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.
- XYZ\(^2\) hexagonal stabilizer code— The XYZ\(^2\) hexagonal stabilizer code can be viewed as a concatenation of the \(YZZY\) surface code with one of the possible \([[2,1]]\) repetition codes, with the case of the bit-flip repetition code yielding a concatenation of the surface code with the dual-rail code [31].
Member of code lists
- 2D stabilizer codes
- Asymmetric quantum codes and friends
- Fracton codes and friends
- Hamiltonian-based codes and friends
- Quantum codes
- Quantum codes with code capacity thresholds
- Quantum codes with notable decoders
- Quantum codes with other thresholds
- Quantum LDPC codes
- Qubit QLDPC codes
- Realized quantum codes
- Stabilizer codes
- Surface code and friends
- Topological codes
Primary Hierarchy
Clifford-deformed surface code (CDSC)Lattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Parents
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 \(XZXZ\).
XZZX toric and planar codes can be treated in the general twist-defect surface code formalism [32].
Abelian quantum-double stabilizer codeLattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
The XZZX surface code is an example of \(\mathbb{Z}_2\) topological order as manifest in the Wen plaquette model [1].
XZZX surface code
Children
Imposing twisted (a.k.a. shifted) boundary conditions on the toric XZZX code yields the twisted XZZX code [2; Exam. 11 and Fig. 3][32; Fig. 6].
References
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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