## Description

Five-qubit stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. Its generators that 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 five-qubit code is the smallest known example of quantum cyclic code. Its automorphism group is the dihedral group of order 10 [3].

## Protection

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

## Encoding

## Transversal Gates

Pauli gates are transversal, along with a non-Pauli Hadamard-phase gate \(SH\) and three-qubit Clifford operation \(M_3\) [6,7]. These realize the \(2T\) binary tetrahedral subgroup of \(SU(2)\). The code does not admit any non-Clifford transversal gates [8]. All such gates can be interpreted as monodromies under a particular notion of parallel transport [9; Exam. 6.4.2].

## Gates

Pieceable fault-tolerant CZ, CNOT, and CCZ gates [10].

## Decoding

Combined dynamical decoupling and error correction protocol on individually-controlled qubits with always-on Ising couplings [5].Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived [11].

## Fault Tolerance

Pieceable fault-tolerant CZ, CNOT, and CCZ gates [10].Syndrome measurement can be done with two ancillary flag qubits [12]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [13].

## Realizations

NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking [14]. Single-qubit logical gates [15].Superconducting qubits [16].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 [17].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations [18].

## Parents

- XZZX surface code — The five-qubit code is the smallest XZZX surface code [19; Ex. 11][20; Ex. 3].
- Perfect quantum code — The five-qubit code is the smallest perfect code.
- Stabilizer code over \(GF(4)\) — The five-qubit code is derived from the \([5,3,3]_4\) shortened hexacode via the stabilizer-over-\(GF(4)\) construction.
- Quantum maximum-distance-separable (MDS) code — The five-qubit code is one of the two qubit quantum MDS codes.
- \([[5,1,3]]_{\mathbb{Z}_q}\) 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.

## Cousins

- Hamiltonian-based code — The five-qubit code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [21].
- Majorana stabilizer code — The five-qubit code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [21].
- Hexacode — Applying the stabilizer-over-\(GF(4)\) construction to the hexacode yields a \([[6,0,4]]\) quantum code [22] corresponding to the six-qubit perfect state. The five-qubit code can be obtained either by applying the stabilizer-over-\(GF(4)\) construction to the shortened hexacode or by tracing out a qubit of the \([[6,0,4]]\) code [23].
- \(U(d)\)-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.
- Quantum divisible code — A fault-tolerant \(T\) gate on the five-qubit code can be obtained by concatenating with particular quantum divisible codes.
- \([[4,2,2]]\) CSS code — \([[4,2,2]]\) 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; see Sec. 3.5 in Gottesman [7].
- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code — The five-qubit encoding isometry tiles various holographic codes because its corresponding tensor is perfect [23].
- \((5,1,2)\)-convolutional code — The \((5,1,2)\)-convolutional code is an infinite-qubit version of the five-qubit perfect code, with the former's lattice-translation symmetry being the extension of the latter's cyclic permutation symmetry.

## 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]
- M. Nakahara, “Quantum Computing”, (2008) DOI
- [5]
- 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
- [6]
- D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
- [7]
- D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
- [8]
- E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
- [9]
- D. Gottesman and L. L. Zhang, “Fibre bundle framework for unitary quantum fault tolerance”, (2017) arXiv:1309.7062
- [10]
- 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
- [11]
- C. Liu, “Exact performance of the five-qubit code with coherent errors”, (2022) arXiv:2203.01706
- [12]
- R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
- [13]
- D. Bhatnagar et al., “Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes”, (2023) arXiv:2305.00784
- [14]
- E. Knill et al., “Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code”, Physical Review Letters 86, 5811 (2001) arXiv:quant-ph/0101034 DOI
- [15]
- 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
- [16]
- 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
- [17]
- C. Ryan-Anderson et al., “Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code”, (2022) arXiv:2208.01863
- [18]
- 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
- [19]
- 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
- [20]
- 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
- [21]
- Aleksander Kubica, private communication, 2019
- [22]
- 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
- [23]
- 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

## 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