Five-qubit perfect code[1,2] 


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].


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 ([4], Fig. 10.16).Four CNOT and five CPHASE gates [5].

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].


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


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].


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].



  • 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.


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“Five-qubit perfect code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
  title={Five-qubit perfect code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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“Five-qubit perfect code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.