\([[4,1,1,2]]\) Four-qubit subsystem code[1,2] 

Description

Error-detecting four-qubit subsystem stabilizer code encoding one logical qubit and one gauge qubit.

The \([[4,1,1,2]]\) code can be obtained by picking one of the logical qubits of the \([[4,2,2]]\) four-qubit code to be a gauge qubit; e.g., see Ref. [3]. One particular gauge configuration has gauge operators \(\{XXII,IIXX,ZIZI,IZIZ\}\).

Parent

  • Bacon-Shor code — The four-qubit subsystem code is the shortest error-detecting Bacon-Shor code.

Cousins

  • \([[4,2,2]]\) Four-qubit code — The \([[4,1,1,2]]\) code can be obtained by picking one of the logical qubits of the \([[4,2,2]]\) four-qubit code to be a gauge qubit; e.g., see Ref. [3]. One particular gauge configuration has gauge operators \(\{XXII,IIXX,ZIZI,IZIZ\}\).
  • 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.
  • Raussendorf-Bravyi-Harrington (RBH) cluster-state code — Concatenation of the RBH code with small codes such as the \([[2,1,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds [4].

References

[1]
P. W. Shor, “Scheme for reducing decoherence in quantum computer memory”, Physical Review A 52, R2493 (1995) DOI
[2]
D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories”, Physical Review A 73, (2006) arXiv:quant-ph/0506023 DOI
[3]
V. V. Albert, Boulder School 2023 Lecture notes
[4]
Z. Li, I. Kim, and P. Hayden, “Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction”, Quantum 7, 1089 (2023) arXiv:2209.09390 DOI
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Zoo Code ID: bacon_shor_4

Cite as:
\([[4,1,1,2]]\) Four-qubit subsystem code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/bacon_shor_4
BibTeX:
@incollection{eczoo_bacon_shor_4, title={\([[4,1,1,2]]\) Four-qubit subsystem code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/bacon_shor_4} }
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https://errorcorrectionzoo.org/c/bacon_shor_4

Cite as:

\([[4,1,1,2]]\) Four-qubit subsystem code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/bacon_shor_4

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/subsystem/bacon_shor_4.yml.