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\([[7,1,3]]\) bare code[1]

Description

A \([[7,1,3]]\) code that admits fault-tolerant syndrome extraction using only one ancilla per stabilizer generator measurement.

A stabilizer tableau for the code is given by [2; ID 108] \begin{align} \begin{array}{ccccccc} X & I & I & I & Z & I & I \\ I & X & I & I & Z & I & I \\ I & I & X & I & I & Z & I \\ I & I & I & X & I & I & Z \\ I & I & Z & Z & I & Y & Y \\ Z & Z & Z & I & X & X & Z \end{array}~. \tag*{(1)}\end{align} It is one of sixteen distinct indecomposable \([[7,1,3]]\) codes [3].

Decoding

Fault-tolerant syndrome extraction using a single ancilla [1].

Cousin

  • \([[7,2,2]]\) QETC— The stabilizer group of the \([[7,2,2]]\) QETC, together with the logical-\(Z\) operator on the first logical qubit, generates the stabilizer group of a \([[7,1,3]]\) code [4] equivalent to the bare \([[7,1,3]]\) code [5].

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
Small-distance qubit stabilizer codeSmall-distance block quantum QECC Quantum
Parents
Small-distance qubit stabilizer code
\([[7,1,3]]\) bare code

References

[1]
M. Li, M. Gutiérrez, S. E. David, A. Hernandez, and K. R. Brown, “Fault tolerance with bare ancillary qubits for a [[7,1,3]] code”, Physical Review A 96, (2017) arXiv:1702.01155 DOI
[2]
Qiskit Community, “Qiskit QEC framework”, URL
[3]
S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
[4]
D. Zhang and T. Cubitt, “Quantum Error Transmutation”, (2023) arXiv:2310.10278
[5]
V. V. Albert, private communication, 2026
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Zoo Code ID: bare_7_1_3

Cite as:
“\([[7,1,3]]\) bare code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/bare_7_1_3
BibTeX:
@incollection{eczoo_bare_7_1_3, title={\([[7,1,3]]\) bare code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/bare_7_1_3} }
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Permanent link:
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Cite as:

“\([[7,1,3]]\) bare code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/bare_7_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/7/bare_7_1_3.yml.