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

Description

One of two six-qubit distance-three codes that are unique up to equivalence [1], with the other code a trivial extension of the five-qubit code [2]. The code admits fault-tolerant syndrome extraction using only one ancilla per stabilizer generator measurement.

Stabilizer generators and logical Pauli operators are presented in Ref. [2,3]. The code is equivalent to the cluster state in Ref. [4].

Encoding

CNOT and Hadamard gates [2,3].

Gates

Logical CNOT gate [2,3].

Decoding

Fault-tolerant syndrome extraction using a single ancilla [4].

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 [2,3].
  • 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], with the other code a trivial extension of the five-qubit code [2].

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]
B. A. Shaw, “Quantum Steganography and Quantum Error-Correction”, (2010) arXiv:1008.0425
[4]
H. Gupta, P. Maheshwari, and A. Raina, “Fault-tolerance of [[6, 1, 3]] non-CSS code family generated using measurements on graph states”, (2025) arXiv:2501.12072
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Zoo Code ID: stab_6_1_3

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
BibTeX:
@incollection{eczoo_stab_6_1_3, title={\([[6,1,3]]\) Six-qubit stabilizer code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_6_1_3} }
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Permanent link:
https://errorcorrectionzoo.org/c/stab_6_1_3

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.