\([[4,2,2]]\) Four-qubit code[1,2] 

Also known as \(C_4\) code.

Description

Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error.

Admits generators \(\{XXXX, ZZZZ\} \) and codewords \begin{align} \begin{split} |\overline{00}\rangle = (|0000\rangle + |1111\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{01}\rangle = (|0011\rangle + |1100\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{10}\rangle = (|0101\rangle + |1010\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{11}\rangle = (|0110\rangle + |1001\rangle)/\sqrt{2}~. \end{split} \tag*{(1)}\end{align} This code is the smallest instance of the toric code, and its various single-qubit subcodes are small planar surface codes. It is the unique code for its parameters, up to equivalence [3; Thm. 8].

The subcode \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) is a \([[4,1,2]]\) code known as the Leung-Nielsen-Chuang-Yamamoto code [4], whose \(\pm\)-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 [1] or single erasure [2]. Not able to correct arbitrary single-qubit errors because \( \lfloor \frac{d-1}{2} \rfloor =0 \).

The \([[4,1,2]]\) subcodes \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [4] and \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [5] approximately correct a single AD error, with the latter being a constant excitation code.

An equivalent version of this code can suppress errors in adiabatic quantum computation by being used as an excited-state space of a particular Hamiltonian [6].

Magic

Various magic-state distillation protocols exist for the \([[4,2,2]]\) qubit code and the \(C_6\) code in what are known as Meier-Eastin-Knill (MEK) protocols [7]. For example, the magic-state yield parameter is \(\gamma = \log_2 5 \approx 2.322\) for a protocol using the \([[10,2,2]]\) code [8; Box 2]; see also [9; Table IV].

Transversal Gates

Transversal Pauli, Hadamard, and two-qubit \(S=\sqrt{Z}\) [10] (see also [11]).A transversal \(CZ\) gate is realized by the rotation \(\sqrt{Z}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}\).Adding \(XYZI\) to the stabilizer group produces a \([[4,1,2]]\) subcode that admits weight-two transversal logical Pauli operations [12].

Gates

Some inter-block gates can be weight-two (two-body) with the help of perturbative gadgets, making it possible to suppress errors in adiabatic quantum computation [6].

Fault Tolerance

Preparation of certain states, both magic and non-magic, along with transversal gates can be performed fault-tolerantly, but requires post-selection because the code cannot correct errors [10]. Magic states can be injected into surface and color codes since the code is a small instance of both [13].Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [14] (see also Ref. [15]).Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [16,17].Fault-tolerant implementation of the Deutsch-Josza algorithm [18].

Realizations

See also [19; Tab. I] for more details on each experimental realization.\([[4,1,2]]\) subcodes implemented in linear optical networks [20,21].Trapped-ion device by IonQ [22].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [23–26].The subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [27] and \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [28], treated as a planar surface code, has been realized in superconducting-circuit devices.Logical gates between two copies of the subcode \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\), interpreted as lattice surgery between planar surface codes, realized in superconducting circuits [29].Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits [30].The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device [13].Logical Clifford gates for a twist-defect surface code that is single-qubit Clifford equivalent to a \([[4,1,2]]\) realized in a trapped ion device by Quantinuum [31].CPC gadgets for the \([[4,2,2]]\) code have been implemented on the IBM 5Q superconducting device [32].An FPGA implementation of the collision clustering decoder [33] realized on a Rigetti supercondutcing device [34].Rydberg atomic devices: error detection, erasure correction, and post-selected fault-tolerant circuits demonstrated on 24 logical qubits on a 256-qubit device by Atom Computing, with each qubit encoded in the \([[4,2,2]]\) code [35]. The device also ran the Bernstein-Vazirani algorithm on up to 28 logical qubits encoded in a \([[4,1,2]]\) subcode [35]. Post-selected fault-tolerant realization of a benchmarking protocol [10], preparation of the ground state of the single-impurity Anderson impurity model, and post-selected fault-tolerant logical Bell-state preparation demonstrated on one copy of the \([[4,2,2]]\) code on a device by Infleqtion [19].

Parents

  • Toric code — The \([[4,2,2]]\) code is the smallest toric code. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [36].
  • 2D color code — The \([[4,2,2]]\) code can be interpreted as a 2D color code on a square of the 4.8.8 or 4.6.12 tilings, or on a trapezoidal patch that makes up two-thirds of a hexagon of the 6.6.6 tiling [13,37]. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [36]. Concatenating the \([[4,2,2]]\) code with two copies of the surface code yields the 4.6.12 color code [36]. A small 6.6.6 color code is a \([[4,1,2]]\) subcode with three weight-three stabilizer generators [38; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [31] via a single-qubit Clifford circuit.
  • \([[2^D,D,2]]\) hypercube quantum code — The \([[4,2,2]]\) code is a hypercube code for \(D=2\).
  • \([[2m,2m-2,2]]\) error-detecting code — The \([[2m,2m-2,2]]\) error-detecting code for \(m=2\) reduces to the \([[4,2,2]]\) code.
  • \([[4,2,2]]_{G}\) four group-qudit code — The four group-qudit code reduces to the four-rotor code for \(G= \mathbb{Z}_2\).
  • Quantum polar code — \([[4,2,2]]\) code is a small quantum polar code [39].

Cousins

  • Twist-defect surface code — A small 6.6.6 color code is a \([[4,1,2]]\) subcode with three weight-three stabilizer generators [38; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [31] via a single-qubit Clifford circuit.
  • Rotated surface code — The subcodes \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\) [29], \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [27], \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [28], and \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) [30] of the \([[4,2,2]]\) code are small planar rotated surface codes.
  • Quantum parity code (QPC) — The \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \([[4,1,2]]\) subcode 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 the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [5,40,41].
  • Five-qubit perfect code — The \([[4,2,2]]\) code can be derived from the five-qubit code using a protocol that converts an \([[n,k,d]]\) code into an \([[n-1, k+1, d-1]]\) code [42; Sec. 3.5].
  • Amplitude-damping (AD) code — The \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) (approximately) corrects a single AD error [4].
  • Binomial code — \([[4,1,2]]\) subcode consisting of \(|\overline{00}\rangle\) and any other codeword 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 [43].
  • Heavy-hexagon code — Magic states prepared using the \([[4,1,2]]\) subcode can be injected into the heavy-hex code [13,26].
  • Concatenated qubit code — The \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \([[4,1,2]]\) subcode is the smallest QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip repetition code. Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [14] (see also Ref. [15]). Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [16,17].' Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [36]. The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code [36]. An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [5,40,41]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [44]. Recursively concatenating a \([[4,1,2]]\) subcode attains a threshold [45,46].
  • Codeword stabilized (CWS) code — A \([[4,1,2]]\) subcode can be thought of as a CWS code [47].
  • \([[7,1,3]]\) Steane code — The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [48].
  • \([[6,4,2]]\) error-detecting code — The \([[6,4,2]]\) error-detecting code can be constructed out of two \([[4,2,2]]\) codes in the quantum Lego code framework [48].
  • Coherent-parity-check (CPC) code — CPC gadgets for the \([[4,2,2]]\) code have been implemented on the IBM 5Q superconducting device [32].
  • Dual-rail quantum code — An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [5,40,41]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [44].
  • Concatenated GKP code — Recursively concatenating the \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [49].
  • Tensor-network code — The Steane and \([[6,4,2]]\) error-detecting codes can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [48].
  • Ladder Floquet code — The smallest example of the ladder Floquet code is a dynamical version of the \([[4,2,2]]\) code [50].
  • Jump code — The subcode \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [5] is a \(((4,2,1))_2\) jump code correcting a single AD error. This code can be extended to a \(((4,3,1))_2\) jump code \(\{|\overline{01}\rangle,|\overline{10}\rangle,|\overline{11}\rangle\}\) [51].
  • Hybrid stabilizer code — The \([[4,2,2]]\) codewords can be modified by signs to yield a \([[4,1:1,2]]\) hybrid stabilizer code [52].
  • Four-qubit single-deletion code — A basis of codewords for the four-qubit single-deletion code consists of the \(|\overline{00}\rangle\) and \(|\overline{01}\rangle+|\overline{10}\rangle+|\overline{11}\rangle\)states of the four-qubit code.
  • Numerically optimized four-qubit AD code — The numerically optimized four-qubit AD code that can correct a single AD error with higher entanglement fidelity than the \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto code subcode of the \([[4,2,2]]\) code.
  • \([[6,2,2]]\) \(C_6\) code — Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [14] (see also Ref. [15]) and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [7]. Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [16,17].
  • \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [16].
  • \([[4,1,1,2]]\) Four-qubit subsystem code — The \([[4,1,1,2]]\) code can be obtained by picking one of the logical qubits of the \([[4,2,2]]\) four-qubit code to be a gauge qubit; e.g., see Ref. [50]. One particular gauge configuration has gauge operators \(\{XXII,IIXX,ZIZI,IZIZ\}\).
  • Heavy-hexagon code — The \(d=2\) heavy-hexagonal code is closely related to the \([[4,1,2]]\) subcode.

References

[1]
L. Vaidman, L. Goldenberg, and S. Wiesner, “Error prevention scheme with four particles”, Physical Review A 54, R1745 (1996) arXiv:quant-ph/9603031 DOI
[2]
M. Grassl, Th. Beth, and T. Pellizzari, “Codes for the quantum erasure channel”, Physical Review A 56, 33 (1997) arXiv:quant-ph/9610042 DOI
[3]
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[4]
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
[5]
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
[6]
Y. Cao, S. Liu, H. Deng, Z. Xia, X. Wu, and Y.-X. Wang, “Robust analog quantum simulators by quantum error-detecting codes”, (2024) arXiv:2412.07764
[7]
A. M. Meier, B. Eastin, and E. Knill, “Magic-state distillation with the four-qubit code”, (2012) arXiv:1204.4221
[8]
E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017) arXiv:1612.07330 DOI
[9]
Quantum Information and Computation 18, (2018) arXiv:1709.02789 DOI
[10]
D. Gottesman, “Quantum fault tolerance in small experiments”, (2016) arXiv:1610.03507
[11]
H. Chen, M. Vasmer, N. P. Breuckmann, and E. Grant, “Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))”, Quantum Information and Computation 22, 947 (2022) arXiv:1912.10063 DOI
[12]
S. P. Jordan, E. Farhi, and P. W. Shor, “Error-correcting codes for adiabatic quantum computation”, Physical Review A 74, (2006) arXiv:quant-ph/0512170 DOI
[13]
R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, Nature 625, 259 (2024) arXiv:2305.13581 DOI
[14]
E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
[15]
J. Cho, “Fault-tolerant linear optics quantum computation by error-detecting quantum state transfer”, Physical Review A 76, (2007) arXiv:quant-ph/0612073 DOI
[16]
H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
[17]
S. Yoshida, S. Tamiya, and H. Yamasaki, “Concatenate codes, save qubits”, (2024) arXiv:2402.09606
[18]
D. Singh and S. Prakash, “Fault-Tolerant Implementation of the Deutsch-Jozsa Algorithm”, (2024) arXiv:2412.04791
[19]
Matt. J. Bedalov et al., “Fault-Tolerant Operation and Materials Science with Neutral Atom Logical Qubits”, (2024) arXiv:2412.07670
[20]
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
[21]
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
[22]
N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, “Fault-tolerant quantum error detection”, Science Advances 3, (2017) arXiv:1611.06946 DOI
[23]
M. Takita, A. W. Cross, A. D. Córcoles, J. M. Chow, and J. M. Gambetta, “Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits”, Physical Review Letters 119, (2017) arXiv:1705.09259 DOI
[24]
Quantum Information and Computation 18, (2018) arXiv:1705.08957 DOI
[25]
R. Harper and S. T. Flammia, “Fault-Tolerant Logical Gates in the IBM Quantum Experience”, Physical Review Letters 122, (2019) arXiv:1806.02359 DOI
[26]
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
[27]
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
[28]
“Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
[29]
A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 DOI
[30]
J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI
[31]
S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
[32]
J. Roffe, D. Headley, N. Chancellor, D. Horsman, and V. Kendon, “Protecting quantum memories using coherent parity check codes”, Quantum Science and Technology 3, 035010 (2018) arXiv:1709.01866 DOI
[33]
B. Barber et al., “A real-time, scalable, fast and highly resource efficient decoder for a quantum computer”, (2024) arXiv:2309.05558
[34]
L. Caune et al., “Demonstrating real-time and low-latency quantum error correction with superconducting qubits”, (2024) arXiv:2410.05202
[35]
B. W. Reichardt et al., “Logical computation demonstrated with a neutral atom quantum processor”, (2024) arXiv:2411.11822
[36]
B. Criger and B. Terhal, “Noise thresholds for the [4,2,2]-concatenated toric code”, Quantum Information and Computation 16, 1261 (2016) arXiv:1604.04062 DOI
[37]
M. S. Kesselring, J. C. Magdalena de la Fuente, F. Thomsen, J. Eisert, S. D. Bartlett, and B. J. Brown, “Anyon Condensation and the Color Code”, PRX Quantum 5, (2024) arXiv:2212.00042 DOI
[38]
M. S. Kesselring, F. Pastawski, J. Eisert, and B. J. Brown, “The boundaries and twist defects of the color code and their applications to topological quantum computation”, Quantum 2, 101 (2018) arXiv:1806.02820 DOI
[39]
Kyungjoo Noh, Leung code as quantum polar code, 2017.
[40]
T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
[41]
Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
[42]
D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
[43]
Linshu Li, private communication, 2018
[44]
R. Duan, M. Grassl, Z. Ji, and B. Zeng, “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory (2010) arXiv:1001.2356 DOI
[45]
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
[46]
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
[47]
C. Cafaro, D. Markham, and P. van Loock, “Scheme for constructing graphs associated with stabilizer quantum codes”, (2014) arXiv:1407.2777
[48]
C. Cao and B. Lackey, “Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks”, PRX Quantum 3, (2022) arXiv:2109.08158 DOI
[49]
K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
[50]
V. V. Albert, Boulder School 2023 Lecture notes
[51]
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
[52]
S. Majidy, “A Unification of the Coding Theory and OAQEC Perspectives on Hybrid Codes”, International Journal of Theoretical Physics 62, (2023) arXiv:1806.03702 DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: stab_4_2_2

Cite as:
“\([[4,2,2]]\) Four-qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_4_2_2
BibTeX:
@incollection{eczoo_stab_4_2_2, title={\([[4,2,2]]\) Four-qubit code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_4_2_2} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/stab_4_2_2

Cite as:

“\([[4,2,2]]\) Four-qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_4_2_2

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/4/stab_4_2_2.yml.