## Description

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].

## 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 \(M_3\) [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].

## Gates

## Decoding

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].

## Realizations

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.

## Parents

- XZZX surface code — The five-qubit code is the smallest XZZX surface code [24; Ex. 11][25; Ex. 3].
- 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 [26][27; Exam. A].
- 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.
- Group-representation code — The five-qubit code is a group-representation code with \(G\) being the \(2T\) subgroup of \(SU(2)\) [28].

## Cousins

- Hamiltonian-based code — The five-qubit code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [29].
- Majorana stabilizer code — The five-qubit code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [29].
- Qubit code — Every \(((5,2,3))\) code is equivalent to the five-qubit code [4; Corr. 10].
- Concatenated quantum code — The concatenated five-qubit code has a measurement threshold of one [30].
- Hexacode — Applying the qubit Hermitian construction to the hexacode yields a \([[6,0,4]]\) quantum code [26] corresponding to the six-qubit perfect state. The five-qubit code can be obtained either by applying the qubit Hermitian construction to the shortened hexacode [27; Exam. A] or by tracing out a qubit of the \([[6,0,4]]\) code [31; Appx. A].
- \(U(d)\)-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.
- \([[4,2,2]]\) CSS 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; see Sec. 3.5 in Gottesman [10].
- \((5,1,2)\)-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.
- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code — The five-qubit encoding isometry tiles various holographic codes because its corresponding tensor is perfect [31].
- Quantum divisible code — A fault-tolerant \(T\) gate on the five-qubit code can be obtained by concatenating with particular quantum divisible codes [32].

## References

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- 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