Five-qubit perfect code

Five-qubit perfect code[1,2]

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].Fault-tolerant logical one and logical minus state preparation in all-to-all and 2D grid connectivity [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

Magic-state distillation protocol [14].Pieceable fault-tolerant CZ, CNOT, and CCZ gates [15].

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].Fault-tolerant logical one and logical minus state preparation in all-to-all and 2D grid connectivity [9].

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 qubit code — The recursively 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

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Remmy Zen (2024-07-15) — most recent
Victor V. Albert (2022-08-10)
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.), 2024. https://errorcorrectionzoo.org/c/stab_5_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/stab_5_1_3.yml.