Five-qubit perfect code[1,2] 


Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error.

Its generators are symmetric under cyclic permutation of qubits, \begin{align} \begin{split} S_1 &= IXZZX \\ S_2 &= XZZXI \\ S_3 &= ZZXIX \\ S_4 &= ZXIXZ~. \end{split} \tag*{(1)}\end{align} The code's automorphism group is the dihedral group of order 10 [3].

It is the unique code for its parameters, up to local equivalence [4; Corr. 10]. In fact, any 5 qubit \(2T\)-transversal stabilizer code with distance \(d>1\) must be the five-qubit code [5,6].


Smallest stabilizer code that protects against a single error on any one qubit. Detects two-qubit errors.


Four CCZ, four Hadamard, and one \(Z\) gate ([7], Fig. 10.16).Four CNOT and five CPHASE gates [8].

Transversal Gates

Pauli gates are transversal, along with a non-Pauli Hadamard-phase gate \(SH\) and three-qubit Clifford operation \(M_3\) [9,10]. These realize the \(2T\) binary tetrahedral subgroup of \(SU(2)\). The code does not admit any non-Clifford transversal gates [4]. All such gates can be interpreted as monodromies under a particular notion of parallel transport [11; Exam. 6.4.2].


Magic-state distillation protocol [12].Pieceable fault-tolerant CZ, CNOT, and CCZ gates [13].


Combined dynamical decoupling and error correction protocol on individually-controlled qubits with always-on Ising couplings [8].Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived [14].

Fault Tolerance

Pieceable fault-tolerant CZ, CNOT, and CCZ gates [13].Syndrome measurement can be done with two ancillary flag qubits [15]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [16].


NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking [17]. Single-qubit logical gates [18]. Magic-state distillation using 7-qubit device [19].Superconducting qubits [20].Trapped-ion qubits: non-transversal CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 12-qubit device by Quantinuum [21]. Real-time magic-state distillation [22].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations using two ancillas [23]. The fault-tolerant circuit yields better fidelity than the non-fault-tolerant circuit.




R. Laflamme et al., “Perfect Quantum Error Correction Code”, (1996) arXiv:quant-ph/9602019
C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
H. Hao, “Investigations on Automorphism Groups of Quantum Stabilizer Codes”, (2021) arXiv:2109.12735
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
E. Kubischta, I. Teixeira, and J. M. Silvester, “Quantum Weight Enumerators for Real Codes with \(X\) and \(Z\) Exactly Transversal”, (2024) arXiv:2306.12526
Ian Teixeira, private communication, 2024
M. Nakahara, “Quantum Computing”, (2008) DOI
A. De and L. P. Pryadko, “Universal set of dynamically protected gates for bipartite qubit networks: Soft pulse implementation of the [[5,1,3]] quantum error-correcting code”, Physical Review A 93, (2016) arXiv:1509.01239 DOI
D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
D. Gottesman and L. L. Zhang, “Fibre bundle framework for unitary quantum fault tolerance”, (2017) arXiv:1309.7062
S. Bravyi and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas”, Physical Review A 71, (2005) arXiv:quant-ph/0403025 DOI
T. J. Yoder, R. Takagi, and I. L. Chuang, “Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes”, Physical Review X 6, (2016) arXiv:1603.03948 DOI
C. Liu, “Exact performance of the five-qubit code with coherent errors”, (2022) arXiv:2203.01706
R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
D. Bhatnagar et al., “Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes”, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) (2023) arXiv:2305.00784 DOI
E. Knill et al., “Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code”, Physical Review Letters 86, 5811 (2001) arXiv:quant-ph/0101034 DOI
J. Zhang, R. Laflamme, and D. Suter, “Experimental Implementation of Encoded Logical Qubit Operations in a Perfect Quantum Error Correcting Code”, Physical Review Letters 109, (2012) arXiv:1208.4797 DOI
A. M. Souza et al., “Experimental magic state distillation for fault-tolerant quantum computing”, Nature Communications 2, (2011) arXiv:1103.2178 DOI
M. Gong et al., “Experimental exploration of five-qubit quantum error-correcting code with superconducting qubits”, National Science Review 9, (2021) arXiv:1907.04507 DOI
C. Ryan-Anderson et al., “Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code”, (2022) arXiv:2208.01863
N. C. Brown et al., “Advances in compilation for quantum hardware -- A demonstration of magic state distillation and repeat-until-success protocols”, (2023) arXiv:2310.12106
M. H. Abobeih et al., “Fault-tolerant operation of a logical qubit in a diamond quantum processor”, Nature 606, 884 (2022) arXiv:2108.01646 DOI
A. A. Kovalev, I. Dumer, and L. P. Pryadko, “Design of additive quantum codes via the code-word-stabilized framework”, Physical Review A 84, (2011) arXiv:1108.5490 DOI
A. A. Kovalev and L. P. Pryadko, “Quantum Kronecker sum-product low-density parity-check codes with finite rate”, Physical Review A 88, (2013) arXiv:1212.6703 DOI
A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
G. D. Forney, M. Grassl, and S. Guha, “Convolutional and Tail-Biting Quantum Error-Correcting Codes”, IEEE Transactions on Information Theory 53, 865 (2007) arXiv:quant-ph/0511016 DOI
A. Denys and A. Leverrier, “Multimode bosonic cat codes with an easily implementable universal gate set”, (2023) arXiv:2306.11621
Aleksander Kubica, private communication, 2019
D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015) arXiv:1503.06237 DOI
J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
Page edit log

Your contribution is welcome!

on (edit & pull request)— see instructions

edit on this site

Zoo Code ID: stab_5_1_3

Cite as:
“Five-qubit perfect code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_stab_5_1_3, title={Five-qubit perfect code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Five-qubit perfect code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.