\([[5,1,3]]\) perfect code

\([[5,1,3]]\) perfect code[1]

Description

Five-qubit stabilizer code with 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} \end{align}

Protection

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

Encoding

Four CNOT and five CPHASE gates [2].

Transversal Gates

Pauli gates are transversal.

Gates

Pieceable fault-tolerant CZ and CCZ gates [3].

Decoding

Combined dynamical decoupling and error correction protocol on individually-controlled qubits with always-on Ising couplings [2].Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived [4].

Fault Tolerance

Pieceable fault-tolerant CZ and CCZ gates [3].Syndrome measurement can be done with two ancillary flag qubits [5].

Realizations

First realized in NMR [6].Demonstration with superconducting qubits [7].

Parent

Stabilizer code over \(GF(4)\) — The \([[5,1,3]]\) code is derived from the \([5,3,3]_4\) Hamming code.

Cousins

Perfect quantum code — The smallest perfect code.
Quantum maximum-distance-separable (MDS) code — The smallest quantum MDS code.
Quantum cyclic code — \([[5,1,3]]\) code is the smallest known example of quantum cyclic code .
Hamiltonian-based code — \([[5,1,3]]\) code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [8].
Majorana stabilizer code — \([[5,1,3]]\) code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [8].
Pastawski-Yoshida-Harlow-Preskill (HaPPY) code — The \([[5,1,3]]\) encoding isometry tiles various holographic codes because its corresponding tensor is perfect [9].
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 \([[5,1,3]]\) code using a protocal that converts \([[n,k,d]]\) code into an \([[n-1, k+1, d-1]]\) code; see Sec. 3.5 in Gottesman [10].

References

[1]: Raymond Laflamme et al., “Perfect Quantum Error Correction Code”. quant-ph/9602019
[2]: 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). DOI; 1509.01239
[3]: T. J. Yoder, R. Takagi, and I. L. Chuang, “Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes”, Physical Review X 6, (2016). DOI; 1603.03948
[4]: Chaobin Liu, “Exact performance of the five-qubit code with coherent errors”. 2203.01706
[5]: R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018). DOI; 1705.02329
[6]: E. Knill et al., “Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code”, Physical Review Letters 86, 5811 (2001). DOI
[7]: M. Gong et al., “Experimental exploration of five-qubit quantum error-correcting code with superconducting qubits”, National Science Review 9, (2021). DOI; 1907.04507
[8]: Aleksander Kubica, private communication, 2019
[9]: F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015). DOI; 1503.06237
[10]: Daniel Gottesman, “Stabilizer Codes and Quantum Error Correction”. quant-ph/9705052

Cite as:

“\([[5,1,3]]\) perfect code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_5_1_3

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