Five-qubit perfect code

Five-qubit perfect code[1,2]

Alternative names: 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]. The encoder-respecting form of the code is a pentagon graph with an additional central input node [4].

It is the unique code for its parameters, up to equivalence [5; Corr. 10]. Any 5 qubit \(2T\)-transversal stabilizer code with distance \(d>1\) must be the five-qubit code [6,7].

This code is sometimes referred to as the DiVincenzo-Shor code after a paper that studied the code's syndrome extraction circuits [8]

Protection

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

Encoding

Nine single- and two-qubit unitaries, six of which are CNOT gates [9].Four generalized control gates, four Hadamard, and one \(Z\) gate [10; Fig. 10.16].Evolution under stabilizer Hamiltonian [11].Four CNOT and five CPHASE gates [12].Reinforcement learning encoding circuits [13].Fault-tolerant logical one and logical minus state preparation in all-to-all and 2D grid connectivity [13].

Transversal Gates

Pauli gates are transversal, along with a non-Pauli Hadamard-phase "facet" gate \(SH\) and three-qubit Clifford operation \(M_3\) [14,15]. These realize the \(2T\) binary tetrahedral subgroup of \(SU(2)\).The entire logical Clifford group can be realized using fold-transversal gates [16–18].The code does not admit any non-Clifford transversal gates [5]; in particular, see [19] for the case of collective \(Z\) rotations.Transversal gates can be interpreted as monodromies under a particular notion of parallel transport [20; Exam. 6.4.2].

Gates

Magic-state distillation protocol [21].Pieceable fault-tolerant CZ, CNOT, and \(CCZ\) gates [16].

Decoding

Fault-tolerant syndrome extraction circuits [8,22].Syndrome extraction circuit optimized for a linear qubit architecture [23].Combined dynamical decoupling and error correction protocol on individually-controlled qubits with always-on Ising couplings [12].Syndrome extraction circuit using only CNOT-SWAP gates [24].Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived [25].Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [26].

Fault Tolerance

Pieceable fault-tolerant CZ, CNOT, and \(CCZ\) gates [16].Syndrome measurement can be done with two ancillary flag qubits [27]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [28].Fault-tolerant logical one and logical minus state preparation in all-to-all and 2D grid connectivity [13].Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [26].

Threshold

Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [29].

Realizations

NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking [30]. Single-qubit logical gates [31]. Magic-state distillation using 7-qubit device [32].Superconducting qubits [33].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 [34]. Real-time magic-state distillation [35].Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations using two ancillas [36]. The fault-tolerant circuit yields better fidelity than the non-fault-tolerant circuit.

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 [5; 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][45; Table I].
Hastings-Haah Floquet code— Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed for the five-qubit code [26].
Hexacode— Applying the qubit Hermitian construction to the hexacode yields a \([[6,0,4]]\) quantum code [46] 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 [47; Exam. A] or by tracing out a qubit of the \([[6,0,4]]\) code [48; 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 [49]. The former code can be obtained by converting the latter into a code that is oblivious to collective \(Z\)-type rotations [19; Exam. 6].
Concatenated GKP code— GKP codes have been concatenated with the five-qubit code [50].
\(U(d)\)-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.
Constant-excitation (CE) code— The five-qubit code can be concatenated with a particular decoherence-free subspace (DFS) [51–54] to yield a 20-qubit CE code [55,56].
\([[3, 1, 3;2]]\) EA code— The \([[3, 1, 3;2]]\) EA code and the five-qubit code have the same stabilizers [57,58].
\([[4,2,2]]\) 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 [15; Sec. 3.5][59; Fig. 3].
\([[6,1,3]]\) 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 [60], with the other code a trivial extension of the five-qubit code [61].
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 [62].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Twist-defect surface codeLattice stabilizer QLDPC Stabilizer Hamiltonian-based QECC Quantum

XZZX surface codeCDSC Qubit Lattice stabilizer QLDPC Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Twisted XZZX toric codeCyclic quantum QECC Quantum

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 [45; Exam. 11 and Fig. 3][63; Exam. 3][64; Fig. 1]. Doubling the five-qubit code via the formalism of Ref. [64] yields a \([[10,2,3]]\) code.

\((5,1,2)\)-convolutional codeLattice stabilizer QLDPC Stabilizer Hamiltonian-based Qubit QECC Quantum

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. [65] for the first few codes in a different extension of the five-qubit perfect code.

Pastawski-Yoshida-Harlow-Preskill (HaPPY) codeStabilizer Hamiltonian-based Qubit HQECC QECC Quantum

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 [48].

Perfect quantum codeSmall-distance block quantum QECC Quantum

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 codeStabilizer Hamiltonian-based Qubit QECC Quantum

The five-qubit code is derived from the \([5,3,3]_4\) shortened hexacode via the qubit Hermitian construction [46][47; Exam. A].

Quantum maximum-distance-separable (MDS) codeQECC Quantum

The five-qubit code is one of the two qubit quantum MDS codes.

Frobenius codeStabilizer Hamiltonian-based Cyclic quantum QECC Quantum

The \([[5,1,3]]\) code is the smallest qubit Frobenius code [66; Table I].

\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit codeStabilizer Hamiltonian-based Cyclic quantum Small-distance block quantum QECC Quantum

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 codeStabilizer Hamiltonian-based Cyclic quantum Small-distance block quantum QECC Quantum

The \([[5,1,3]]_q\) Galois-qudit code for \(q=2\) reduces to the five-qubit perfect code.

Group-representation codeQECC Quantum

The five-qubit code is a group-representation code with \(G\) being the \(2T\) subgroup of \(SU(2)\) [67].

Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum