$[[4,1,2]]$ Leung-Nielsen-Chuang-Yamamoto (LNCY) code

$[[4,1,2]]$ Leung-Nielsen-Chuang-Yamamoto (LNCY) code [1]

Alternative Names: $[[4,1,2]]$ Leung code.

Description

A four-qubit CSS stabilizer code that is the only qubit CSS code with such parameters.

A stabilizer tableau for the code is given by [2; ID 6] \begin{align} \begin{array}{cccc} X & X & I & I \\ I & I & X & X \\ Z & Z & Z & Z \end{array}~.\tag*{$(1)$} \end{align}The code is depicted in Fig. I.

Figure I: Stabilizer generators of the $[[4,1,2]]$ LNCY code. The 4 data qubits (circles) are arranged on a $2\times 2$ rotated surface code lattice with open boundaries. The bulk generator is a weight-four (four-body) $Z$ operator and the two boundary generators are weight-two (two-body) $X$ operators. Red regions correspond to $X$ operators while blue regions correspond to $Z$ operators.

The code admits the following basis of codewords, \begin{align} \begin{split} |\overline{0}\rangle = (|0000\rangle + |1111\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{1}\rangle = (|0011\rangle + |1100\rangle)/\sqrt{2}~. \end{split}\tag*{$(2)$} \end{align}It is realized as the $\{|\overline{00}\rangle,|\overline{01}\rangle\}$ subcode of the $[[4,2,2]]$ four-qubit code [1], and the subcodes spanned by $|\overline{00}\rangle$ and any other $[[4,2,2]]$ codeword are equivalent to it. Applying the Pauli string $IXIX$ to its codewords yields an equivalent constant-excitation $[[4,1,2]]$ code.

The code’s $\pm$-basis codewords can be written as \begin{align} |\overline{\pm}\rangle = \frac{1}{2}(|00\rangle \pm |11\rangle)^{\otimes 2}~.\tag*{$(3)$} \end{align}Using the concatenation convention of the Zoo, this code can be viewed as a concatenation of a two-qubit phase-flip repetition code (inner code) with a two-qubit bit-flip repetition code (outer code).

Protection

Detects a single-qubit error or single erasure as a distance-two code. The code also approximately corrects a single AD error, with recovery fidelity $1-5\gamma^2+O(\gamma^3)$ [1]. A complementary-channel analysis shows that the optimal worst-case entanglement-fidelity distance for this code under AD noise is of order $\gamma$, improving over the uncorrected order $\sqrt{\gamma}$, and that no recovery can improve this asymptotic scaling [3]. The $\{|\overline{01}\rangle,|\overline{11}\rangle\}$ $[[4,1,2]]$ subcode [4] also approximately corrects a single AD error, and is a constant-excitation code.

Fault Tolerance

Recursive concatenation of a $[[4,1,2]]$ error-detecting block, together with message passing across concatenation levels, yields fault-tolerant schemes; a one-dimensional nearest-neighbor architecture has a reported accuracy threshold of approximately $10^{-5}$ per elementary operation [5,6].

Realizations

Linear optical networks [7,8].Superconducting-circuit devices [9,10].Logical gates within one block [11] and between two blocks [12], with the latter interpreted as lattice surgery between planar surface codes, were realized in superconducting circuits.Neutral atom arrays by Atom Computing ran the Bernstein-Vazirani algorithm on up to 28 logical qubits [13].Break-even performance has been demonstrated on a superconducting IBM device using the syndrome-based Petz recovery [14].Implementation of a universal logical gate set in superconducting circuits by Origin Quantum Computing [15].

Cousins

$[[4,2,2]]$ Four-qubit code— The $[[4,1,2]]$ LNCY code is obtained as the $\{|\overline{00}\rangle,|\overline{01}\rangle\}$ $[[4,1,2]]$ subcode of the $[[4,2,2]]$ four-qubit code [1]. A $((4,3,1))_2$ jump code is a subcode of the $[[4,2,2]]$ code and contains the $[[4,1,2]]$ LNCY code as a subcode [16].
$[[4,1,2]]$ twist-defect code— Adding $XXII$ ($XYZI$) to the stabilizer group of the $[[4,2,2]]$ code yields the $[[4,1,2]]$ LNCY (twist-defect) code.
Binomial code— The $[[4,1,2]]$ LNCY code reduces to the $0,2,4$ binomial code when the basis labels in each codeword are written as in base-ten. Such a mapping can be generalized [17].
Heavy-hexagon code— Magic states prepared using a $[[4,1,2]]$ subcode can be injected into the heavy-hex code [18,19]. The $d=2$ heavy-hex code is closely related to the $[[4,1,2]]$ LNCY code.
Cluster-state code— A $[[4,1,2]]$ LNCY code can be thought of as a cluster-state code [20].
Quantum polar code— The $[[4,1,2]]$ LNCY code is a small quantum polar encoding [21].
$[4,3,2]$ single parity-check (SPC) code— The $Z$-type stabilizer of the $[[4,1,2]]$ LNCY code is the parity check of the $[4,3,2]$ SPC code.
Numerically optimized four-qubit AD code— The numerically optimized four-qubit AD code can correct a single AD error with higher entanglement fidelity than the $[[4,1,2]]$ LNCY code [1].
$[[8,2,2]]$ projective-plane color code— Up to qubit permutation, the $[[8,2,2]]$ projective-plane color code is the symplectic double of a local-Clifford-equivalent version of the $[[4,1,2]]$ LNCY code. This version is obtained by applying the phase gate $S$ to every qubit, so its stabilizer group is generated by $YYII$, $IIYY$, and $ZZZZ$.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum

Homological code

Kitaev surface codeCDSC Twist-defect surface QLDPC Qubit CSS Generalized homological-product Lattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Rotated surface codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum

Parents

Rotated surface code

The $[[4,1,2]]$ LNCY code is a small planar rotated surface code [9–12].

Quantum parity code (QPC)CSS Stabilizer Hamiltonian-based Qubit Concatenated quantum QECC Quantum

The $[[4,1,2]]$ LNCY code is the smallest QPC, i.e., a concatenation of a two-qubit phase-flip repetition code (inner code) with a two-qubit bit-flip repetition code (outer code), using the concatenation convention of the Zoo. An $[[8,1,2]]$ QPC correcting a single AD error is equivalent to a concatenation of its constant-excitation version with the dual-rail code [4,22,23]. Recursive concatenation of a $[[4,1,2]]$ error-detecting block admits message-passing fault-tolerance schemes [5,6].

$[[2(m+1),m,2]]$ single-loss AD codeQubit CSS Stabilizer Hamiltonian-based Self-complementary qubit Amplitude-damping (AD) Small-distance block quantum QECC AQECC Quantum

The $[[4,1,2]]$ LNCY code (approximately) corrects a single AD error [1] and is the smallest member of the amplitude-damping stabilizer family of Ref. [24].

Jump codeFock-state bosonic Amplitude-damping (AD) AQECC Qubit CE Hamiltonian-based QECC Quantum

The $[[4,1,2]]$ LNCY code [4] is equivalent to a $((4,2,1))_2$ jump code correcting a single AD error. A $((4,3,1))_2$ jump code is a subcode of the $[[4,2,2]]$ code and contains the $[[4,1,2]]$ LNCY code as a subcode [16].

$[[4,1,2]]$ Leung-Nielsen-Chuang-Yamamoto (LNCY) code

References

[1]
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[4]
G. Alber, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, and M. Mussinger, “Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes”, Physical Review Letters 86, 4402 (2001) arXiv:quant-ph/0103042 DOI
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A. M. Stephens and Z. W. E. Evans, “Accuracy threshold for concatenated error detection in one dimension”, Physical Review A 80, (2009) arXiv:0902.2658 DOI
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Z. W. E. Evans and A. M. Stephens, “Optimal correction of concatenated fault-tolerant quantum codes”, Quantum Information Processing 11, 1511 (2011) arXiv:0902.4506 DOI
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C.-Y. Lu, W.-B. Gao, J. Zhang, X.-Q. Zhou, T. Yang, and J.-W. Pan, “Experimental quantum coding against qubit loss error”, Proceedings of the National Academy of Sciences 105, 11050 (2008) arXiv:0804.2268 DOI
[8]
B. A. Bell, D. A. Herrera-Martí, M. S. Tame, D. Markham, W. J. Wadsworth, and J. G. Rarity, “Experimental demonstration of a graph state quantum error-correction code”, Nature Communications 5, (2014) arXiv:1404.5498 DOI
[9]
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“Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
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J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI
[12]
A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 DOI
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B. W. Reichardt et al., “Fault-tolerant quantum computation with a neutral atom processor”, (2025) arXiv:2411.11822
[14]
D. Biswas and P. Mandayam, “Universal syndrome-based recovery for noise-adapted quantum error correction”, (2026) arXiv:2510.08719
[15]
J. Zhang, Z.-Y. Chen, Y.-J. Wang, B.-H. Lu, H.-F. Zhang, J.-N. Li, P. Duan, Y.-C. Wu, and G.-P. Guo, “Demonstrating a universal logical gate set in error-detecting surface codes on a superconducting quantum processor”, (2024) arXiv:2405.09035
[16]
G. Alber, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, and M. Mussinger, “Detected-jump-error-correcting quantum codes, quantum error designs, and quantum computation”, Physical Review A 68, (2003) arXiv:quant-ph/0208140 DOI
[17]
L. Li, private communication, 2018
[18]
E. H. Chen, T. J. Yoder, Y. Kim, N. Sundaresan, S. Srinivasan, M. Li, A. D. Córcoles, A. W. Cross, and M. Takita, “Calibrated Decoders for Experimental Quantum Error Correction”, Physical Review Letters 128, (2022) arXiv:2110.04285 DOI
[19]
R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, Nature 625, 259 (2024) arXiv:2305.13581 DOI
[20]
C. Cafaro, D. Markham, and P. van Loock, “Scheme for constructing graphs associated with stabilizer quantum codes”, (2014) arXiv:1407.2777
[21]
K. Noh, “Leung code as quantum polar code”, 2017 URL
[22]
T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
[23]
Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
[24]
A. S. Fletcher, P. W. Shor, and M. Z. Win, “Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel”, (2007) arXiv:0710.1052

Page edit log

Victor V. Albert (2026-08-24) — most recent
Victor V. Albert (2026-06-08)
Victor V. Albert (2026-04-23)

Cite as:

“$[[4,1,2]]$ Leung-Nielsen-Chuang-Yamamoto (LNCY) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/css_4_1_2, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/4/css_4_1_2/css_4_1_2.yml.

\([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code [1]