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 group \(\mathsf{G}=\langle i,XXII,IIXX,ZIZI,IZIZ \rangle\).
Parent
- Bacon-Shor code — The four-qubit subsystem code is the shortest error-detecting Bacon-Shor code.
Cousins
- Small-distance block quantum code
- \([[4,2,2]]\) 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 \(\{XXII,IIXX,ZIZI,IZIZ\}\).
- \([[6,2,3,2]]\) BBS code — Both the \([[6,2,3,2]]\) BBS code and the four-qubit subsystem code can be used to suppress errors in adiabatic quantum computation [4].
- Quantum-double code — Quantum double code Hamiltonians can be simulated, with the help of perturbation theory and the four-qubit subsystem code, by two-dimensional two-body Hamiltonians with non-commuting terms [5].
- 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 [6].
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. Jiang and E. G. Rieffel, “Non-commuting two-local Hamiltonians for quantum error suppression”, Quantum Information Processing 16, (2017) arXiv:1511.01997 DOI
- [5]
- C. G. Brell, S. T. Flammia, S. D. Bartlett, and A. C. Doherty, “Toric codes and quantum doubles from two-body Hamiltonians”, New Journal of Physics 13, 053039 (2011) arXiv:1011.1942 DOI
- [6]
- Z. Li, I. Kim, and P. Hayden, “Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction”, Quantum 7, 1089 (2023) arXiv:2209.09390 DOI
Page edit log
- Victor V. Albert (2024-07-01) — most recent
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