\([[4,1,2]]\) 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 [2; ID 6].

The code admits stabilizer generators \(\{XXXX,ZZII,IIZZ\}\) and 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*{(1)}\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*{(2)}\\ \end{align} This code can be thought of as a concatenation of a two-qubit bit-flip with a two-qubit phase-flip 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]. The \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) \([[4,1,2]]\) subcode [3] also approximately corrects a single AD error, and is a constant-excitation code.

Realizations

Linear optical networks [4,5].Superconducting-circuit devices [6,7].Logical gates within one block [8] and between two blocks [9], 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 [10].Break-even performance has been demonstrated on a superconducting IBM device using the syndrome-based Petz recovery [11].Implementation of a universal logical gate set in superconducting circuits by Origin Quantum Computing [12].

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 [13].
\([[4,1,2]]\) 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.
Quantum parity code (QPC)— The \([[4,1,2]]\) LNCY code is the smallest QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip repetition code. 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 [3,14,15].
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 [16].
Heavy-hexagon code— Magic states prepared using a \([[4,1,2]]\) subcode can be injected into the heavy-hex code [17,18]. The \(d=2\) heavy-hex code is closely related to the \([[4,1,2]]\) LNCY code.
Concatenated qubit code— The \([[4,1,2]]\) LNCY code is the smallest QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip repetition code. 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 [3,14,15]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [19]. Recursively concatenating a \([[4,1,2]]\) LNCY subcode attains a threshold [20,21].’
Cluster-state code— A \([[4,1,2]]\) LNCY code can be thought of as a cluster-state code [22].
Quantum polar code— The \([[4,1,2]]\) LNCY code is a small quantum polar encoding [23].
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].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS 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 Concatenated quantum Single-shot QECC Quantum

Parents

Rotated surface code

The \([[4,1,2]]\) LNCY code is a small planar rotated surface code [6–9].

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

The \([[4,1,2]]\) LNCY code [3] is 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 [13].

Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

\([[4,1,2]]\) 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]: A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
[3]: 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
[4]: 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
[5]: 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
[6]: 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
[7]: “Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
[8]: J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI
[9]: A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 DOI
[10]: B. W. Reichardt et al., “Fault-tolerant quantum computation with a neutral atom processor”, (2025) arXiv:2411.11822
[11]: D. Biswas and P. Mandayam, “Universal syndrome-based recovery for noise-adapted quantum error correction”, (2025) arXiv:2510.08719
[12]: 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
[13]: 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
[14]: T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
[15]: Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
[16]: L. Li, private communication, 2018.
[17]: 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
[18]: R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, Nature 625, 259 (2024) arXiv:2305.13581 DOI
[19]: R. Duan, M. Grassl, Z. Ji, and B. Zeng, “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory 2672 (2010) arXiv:1001.2356 DOI
[20]: 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
[21]: 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
[22]: C. Cafaro, D. Markham, and P. van Loock, “Scheme for constructing graphs associated with stabilizer quantum codes”, (2014) arXiv:1407.2777
[23]: K. Noh, Leung code as quantum polar code, 2017.
[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 (2026-04-23) — most recent

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

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