Also known as Laflamme code.

## Description

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

Code generators are symmetric under cyclic permutation of qubits, \begin{align} \begin{split} S_1 &= XZZXI \\ S_2 &= IXZZX \\ S_3 &= XIXZZ \\ 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]. Any 5 qubit \(2T\)-transversal stabilizer code with distance \(d>1\) must be the five-qubit code [5,6].

## Protection

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

## Encoding

Four generalized control gates, four Hadamard, and one \(Z\) gate [7; Fig. 10.16].Four CNOT and five CPHASE gates [8].Reinforcement learning encoding circuits [9].

## Transversal Gates

Pauli gates are transversal, along with a non-Pauli Hadamard-phase gate \(SH\) and three-qubit Clifford operation \(M_3\) [10,11]. These realize the \(2T\) binary tetrahedral subgroup of \(SU(2)\).The code does not admit any non-Clifford transversal gates [4]; in particular, see [12] for the case of collective \(Z\) rotations.All transversal gates can be interpreted as monodromies under a particular notion of parallel transport [13; Exam. 6.4.2].

## Gates

## Decoding

Syndrome extraction circuit using only CNOT-SWAP gates [16].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 [17].

## Fault Tolerance

Pieceable fault-tolerant CZ, CNOT, and CCZ gates [15].Syndrome measurement can be done with two ancillary flag qubits [18]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [19].

## Threshold

Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [20].

## Realizations

NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking [21]. Single-qubit logical gates [22]. Magic-state distillation using 7-qubit device [23].Superconducting qubits [24].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 [25]. Real-time magic-state distillation [26].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations using two ancillas [27]. The fault-tolerant circuit yields better fidelity than the non-fault-tolerant circuit.

## Parents

- Twisted XZZX toric code — Twisted XZZX codes are 2D lattice extensions of the five-qubit perfect code. The five-qubit code is a small twisted XZZX toric code [28; Ex. 11 and Fig. 3][29; Ex. 3][30; Fig. 1]. Doubling the five-qubit code via the formalism of Ref. [30] yields a \([[10,2,3]]\) code.
- \((5,1,2)\)-convolutional code — The \((5,1,2)\)-convolutional code is 1D lattice extension of the five-qubit perfect code, with the former's lattice-translation symmetry being the extension of the latter's cyclic permutation symmetry. The \((5,1,2)\)-convolutional code code reduces to the five-qubit code for a five-qubit chain and periodic boundary conditions. See Ref. [31] for the first few codes in a different extension of the five-qubit perfect code.
- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code — The five-qubit code is the smallest (i.e., radius-one) single-qubit HaPPY code. The five-qubit encoding isometry tiles various holographic codes because its corresponding encoding isometry tensor is a perfect tensor [32].
- Perfect quantum code — The five-qubit code is the smallest perfect code and is a member of the perfect qubit code family \([[(4^r-1)/3, (4^r-1)/3 - 2r, 3]]\) for \(r = 2\).
- Hermitian qubit code — The five-qubit code is derived from the \([5,3,3]_4\) shortened hexacode via the qubit Hermitian construction [33][34; Exam. A].
- Quantum maximum-distance-separable (MDS) code — The five-qubit code is one of the two qubit quantum MDS codes.
- Frobenius code — The \([[5,1,3]]\) code is the smallest qubit Frobenius code [35; Table I].
- \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code — The \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code for \(q=2\) reduces to the five-qubit perfect code.
- \([[5,1,3]]_q\) Galois-qudit code — The \([[5,1,3]]_q\) Galois-qudit code for \(q=2\) reduces to the five-qubit perfect code.
- Group-representation code — The five-qubit code is a group-representation code with \(G\) being the \(2T\) subgroup of \(SU(2)\) [36].

## Cousins

- Majorana stabilizer code — The five-qubit code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [37].
- Qubit code — Every \(((5,2,3))\) code is single-qubit-Clifford-equivalent to the five-qubit code [4; Corr. 10].
- Concatenated quantum code — The concatenated five-qubit code has a measurement threshold of one [38]. Code performance against general Pauli channels has been worked out [39,40].
- Cluster-state code — The five-qubit code admits a codeword that is the cluster state of the pentagon graph [41,42].
- Codeword stabilized (CWS) code — The five-qubit perfect code is equivalent via a single-qubit Clifford circuit to a CWS code defined from a five-cycle graph and a classical repetition code [43,44][28; Table I].
- Hexacode — Applying the qubit Hermitian construction to the hexacode yields a \([[6,0,4]]\) quantum code [33] corresponding to the six-qubit AME state. The five-qubit code can be obtained either by applying the qubit Hermitian construction to the shortened hexacode [34; Exam. A] or by tracing out a qubit of the \([[6,0,4]]\) code [32; Appx. A].
- \([[10,1,4]]_{G}\) tenfold code — The \([[10,1,4]]_{G}\) Abelian group code for \(G=\mathbb{Z}_2\) is defined using a graph that is closely related to the \([[5,1,3]]\) five-qudit code [45]. The former code can be obtained by converting the latter into a code that is oblivious to collective \(Z\)-type rotations [12; Exam. 6].
- \(U(d)\)-covariant approximate erasure code — The five-qubit code can be used to construct an approximate code that is also covariant with respect to the unitary group.
- \([[3, 1, 3;2]]\) EA code — The \([[3, 1, 3;2]]\) EA code and the five-qubit code have the same stabilizers [46,47].
- \([[4,2,2]]\) Four-qubit 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 [11; Sec. 3.5].
- \([[6,1,3]]\) Six-qubit stabilizer code — The \([[6,1,3]]\) six-qubit code is one of two six-qubit distance-three codes that are unique up to equivalence [48], with the other code a trivial extension of the five-qubit code [49].
- Holographic hybrid code — The holographic hybrid code is constructed out of alternating isometries of the five-qubit and \([[4,1,1,2]]\) Bacon-Shor codes.
- Quantum divisible code — A fault-tolerant \(T\) gate on the five-qubit code can be obtained by concatenating with particular quantum divisible codes [50].

## References

- [1]
- R. Laflamme et al., “Perfect Quantum Error Correction Code”, (1996) arXiv:quant-ph/9602019
- [2]
- C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
- [3]
- H. Hao, “Investigations on Automorphism Groups of Quantum Stabilizer Codes”, (2021) arXiv:2109.12735
- [4]
- E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
- [5]
- E. Kubischta, I. Teixeira, and J. M. Silvester, “Quantum Weight Enumerators for Real Codes with \(X\) and \(Z\) Exactly Transversal”, (2024) arXiv:2306.12526
- [6]
- Ian Teixeira, private communication, 2024
- [7]
- M. Nakahara, “Quantum Computing”, (2008) DOI
- [8]
- 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
- [9]
- R. Zen et al., “Quantum Circuit Discovery for Fault-Tolerant Logical State Preparation with Reinforcement Learning”, (2024) arXiv:2402.17761
- [10]
- D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
- [11]
- D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
- [12]
- J. Hu et al., “Mitigating Coherent Noise by Balancing Weight-2 Z-Stabilizers”, IEEE Transactions on Information Theory 68, 1795 (2022) arXiv:2011.00197 DOI
- [13]
- D. Gottesman and L. L. Zhang, “Fibre bundle framework for unitary quantum fault tolerance”, (2017) arXiv:1309.7062
- [14]
- 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
- [15]
- 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
- [16]
- A. V. Antipov, E. O. Kiktenko, and A. K. Fedorov, “Realizing a class of stabilizer quantum error correction codes using a single ancilla and circular connectivity”, Physical Review A 107, (2023) arXiv:2207.13356 DOI
- [17]
- C. Liu, “Exact performance of the five-qubit code with coherent errors”, (2022) arXiv:2203.01706
- [18]
- R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
- [19]
- 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
- [20]
- A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
- [21]
- E. Knill et al., “Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code”, Physical Review Letters 86, 5811 (2001) arXiv:quant-ph/0101034 DOI
- [22]
- 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
- [23]
- A. M. Souza et al., “Experimental magic state distillation for fault-tolerant quantum computing”, Nature Communications 2, (2011) arXiv:1103.2178 DOI
- [24]
- 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
- [25]
- C. Ryan-Anderson et al., “Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code”, (2022) arXiv:2208.01863
- [26]
- 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
- [27]
- 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
- [28]
- 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
- [29]
- 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
- [30]
- R. Sarkar and T. J. Yoder, “A graph-based formalism for surface codes and twists”, (2023) arXiv:2101.09349
- [31]
- Ilya. A. Simakov and Ilya. S. Besedin, “Scalable quantum error correction code on a ring topology of qubits”, (2022) arXiv:2211.03094
- [32]
- 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
- [33]
- 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
- [34]
- 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
- [35]
- S. Dutta and P. P. Kurur, “Quantum Cyclic Code of length dividing \(p^{t}+1\)”, (2011) arXiv:1011.5814
- [36]
- A. Denys and A. Leverrier, “Multimode bosonic cat codes with an easily implementable universal gate set”, (2023) arXiv:2306.11621
- [37]
- Aleksander Kubica, private communication, 2019
- [38]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [39]
- B. Rahn, A. C. Doherty, and H. Mabuchi, “Exact and Approximate Performance of Concatenated Quantum Codes”, (2001) arXiv:quant-ph/0111003
- [40]
- B. Rahn, A. C. Doherty, and H. Mabuchi, “Exact performance of concatenated quantum codes”, Physical Review A 66, (2002) arXiv:quant-ph/0206061 DOI
- [41]
- Griffiths, Robert B. "Graph states and graph codes."
- [42]
- Y. Hwang and J. Heo, “On the relation between a graph code and a graph state”, (2015) arXiv:1511.05647
- [43]
- A. Cross et al., “Codeword Stabilized Quantum Codes”, IEEE Transactions on Information Theory 55, 433 (2009) arXiv:0708.1021 DOI
- [44]
- Cross, Andrew William. Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes. Diss. Massachusetts Institute of Technology, 2008.
- [45]
- D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI
- [46]
- T. Brun, I. Devetak, and M.-H. Hsieh, “Correcting Quantum Errors with Entanglement”, Science 314, 436 (2006) arXiv:quant-ph/0610092 DOI
- [47]
- J. Fan et al., “Entanglement-assisted concatenated quantum codes”, Proceedings of the National Academy of Sciences 119, (2022) arXiv:2202.08084 DOI
- [48]
- A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [49]
- B. Shaw et al., “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI
- [50]
- J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176

## Page edit log

- Victor V. Albert (2022-08-10) — most recent
- Aleksander Kubica (2022-03-14)
- Victor V. Albert (2022-03-14)
- Marianna Podzorova (2021-12-13)
- Qingfeng (Kee) Wang (2021-12-07)

## Cite as:

“Five-qubit perfect code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_5_1_3