\([[4,1,2]]\) 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].
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].
\([[8,2,2]]\) hyperbolic color code— Up to qubit permutation, the \([[8,2,2]]\) hyperbolic 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 codeAmplitude-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]: D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, “Approximate quantum error correction can lead to better codes”, Physical Review A 56, 2567 (1997) arXiv:quant-ph/9704002 DOI
[2]: Qiskit Community, “Qiskit QEC framework”, URL
[3]: C. Bény and O. Oreshkov, “General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels”, Physical Review Letters 104, (2010) arXiv:0907.5391 DOI
[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
[5]: 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
[6]: 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
[7]: 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]: C. K. Andersen, A. Remm, S. Lazar, S. Krinner, N. Lacroix, G. J. Norris, M. Gabureac, C. Eichler, and A. Wallraff, “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020) arXiv:1912.09410 DOI
[10]: “Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
[11]: 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
[13]: 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”, (2025) 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.