\([[6,1,3]]\) Six-qubit stabilizer code

\([[6,1,3]]\) Six-qubit stabilizer code[1,2]

Description

A degenerate, non-trivial \([[6,1,3]]\) stabilizer code. It is one of two six-qubit distance-three codes that are unique up to equivalence [1], with the other code being decomposable and an extension of the five-qubit code [2][3; ID 87]. The code admits fault-tolerant syndrome extraction using only one ancilla per stabilizer generator measurement.

A stabilizer tableau for the code is given by [3; ID 68] \begin{align} \begin{array}{cccccc} X & I & I & I & I & Z \\ I & X & X & Z & Z & I \\ I & Y & I & Y & Z & Z \\ I & X & Z & I & X & Z \\ Z & Y & Z & Z & I & Y \end{array}~. \tag*{(1)}\end{align}

Stabilizer generators and logical Pauli operators are presented in Refs. [2,4]. The code is equivalent to the graph code in Ref. [5].

Encoding

CNOT and Hadamard gates [2,4].

Gates

Logical CNOT gate [2,4].

Decoding

Fault-tolerant syndrome extraction using a single ancilla [5].

Cousins

Subsystem qubit stabilizer code— The \([[6,1,3]]\) six-qubit code can be converted into a \([[6,1,1,3]]\) subsystem code that saturates the subsystem Singleton bound while requiring only four stabilizer measurements during recovery [2,4].
\([[5,1,3]]\) Five-qubit perfect code— The \([[6,1,3]]\) six-qubit code is one of two six-qubit distance-three codes that are unique up to equivalence [1][3; ID 68], with the other code being decomposable and an extension of the five-qubit code [2][3; ID 87].

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

Six-qubit-tensor holographic codeHQECC QECC Quantum

Parents

Six-qubit-tensor holographic code

The \([[6,1,3]]\) six-qubit stabilizer code is the smallest six-qubit-tensor holographic code. The encoding of more general SCF holographic codes is a holographic tensor network consisting of the encoding isometry for the \([[6,1,3]]\) six-qubit stabilizer code.

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

\([[6,1,3]]\) Six-qubit stabilizer code

References

[1]: A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
[2]: B. Shaw, M. M. Wilde, O. Oreshkov, I. Kremsky, and D. A. Lidar, “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI
[3]: Qiskit Community. Qiskit QEC framework. https://github.com/qiskit-community/qiskit-qec
[4]: B. A. Shaw, “Quantum Steganography and Quantum Error-Correction”, (2010) arXiv:1008.0425
[5]: H. Gupta, M. Bhattacharyya, R. Jain, and A. Raina, “Fault-tolerant syndrome extraction in [[n,1,3]] non-CSS code family generated using measurements on graph states”, (2025) arXiv:2501.12072

Page edit log

Victor V. Albert (2024-05-26) — most recent
Matthew Steinberg (2024-05-26)

Cite as:

“\([[6,1,3]]\) Six-qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_6_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/6/stab_6_1_3.yml.