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XZZX surface code[1–4]

Alternative Names: Wen plaquette model.

Description

A variant of the rotated surface code whose generators are \(XZZX\) Pauli strings associated, clockwise, 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; these need not be equivalent to twisted toric codes because they can be non-CSS. 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.

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

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

  • 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].
  • 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].
  • Kitaev surface code— The XZZX surface code on a square lattice with non-twisted periodic boundary conditions is obtained from a surface code by applying Hadamard gates on a subset of qubits such that \(XXXX\) and \(ZZZZ\) generators are both mapped to \(XZXZ\). While this code is equivalent to a CSS surface code with the same distance, other properties like noise-bias performance can differ significantly. Twisted XZZX surface codes are generally not equivalent to CSS surface codes via a single-qubit Clifford circuit and permutation.
  • 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].

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