Bacon–Shor code explained

The Bacon–Shor code is a subsystem error correcting code.^[1] In a subsystem code, information is encoded in a subsystem of a Hilbert space. Subsystem codes lend to simplified error correcting procedures unlike codes which encode information in the subspace of a Hilbert space.^[2] This simplicity led to the first claim of fault tolerant circuit demonstration on a quantum computer.^[3] It is named after Dave Bacon and Peter Shor.

Given the stabilizer generators of Shor's code:

\langle X _{0} X _{1} X _{2} X _{3} X _{4} X _{5} , X _{0} X _{1} X _{2} X _{6} X _{7} X _{8} , Z _{0} Z _{1} , Z _{1} Z _{2} , Z _{3} Z _{4} , Z _{4} Z _{5} , Z _{6} Z _{7} , Z _{7} Z _{8} \rangle , 4 stabilizers can be removed from this generator by recognizing gauge symmetries in the code to get: \langle X _{0} X _{1} X _{2} X _{3} X _{4} X _{5} , X _{0} X _{1} X _{2} X _{6} X _{7} X _{8} , Z _{0} Z _{1} Z _{3} Z _{4} Z _{6} Z _{7} , Z _{1} Z _{2} Z _{4} Z _{5} Z _{7} Z _{8} \rangle .^[4] Error correction is now simplified because 4 stabilizers are needed to measure errors instead of 8. A gauge group can be created from the stabilizer generators:\langle Z _{1} Z _{2} , X _{2} X _{8} , Z _{4} Z _{5} , X _{5} X _{8} , Z _{0} Z _{1} , X _{0} X _{6} , Z _{3} Z _{4} , X _{3} X _{6} , X _{1} X _{7} , X _{4} X _{7} , Z _{6} Z _{7} , Z _{7} Z _{8} \rangle .^[4] Given that the Bacon–Shor code is defined on a square lattice where the qubits are placed on the vertices; laying the qubits on a grid in a way that corresponds to the gauge group shows how only 2 qubit nearest-neighbor measurements are needed to infer the error syndromes. The simplicity of deducing the syndromes reduces the overheard for fault tolerant error correction.^[5]

    ZZ   ZZ 
  q0---q1--q2
XX|  XX|   |XX
  |  ZZ| ZZ|
  q6--q7--q8
XX|  XX|   |XX
  |    |   |
  q3--q4--q5
   ZZ   ZZ

See also

• Five-qubit error correcting code

Notes and References

1. Bacon. Dave. 2006-01-30. Operator quantum error-correcting subsystems for self-correcting quantum memories. Physical Review A. 73. 1. 012340. 10.1103/PhysRevA.73.012340. quant-ph/0506023. 2006PhRvA..73a2340B. 118968017.
2. Book: Aly Salah A., Klappenecker, Andreas . 2008 IEEE International Symposium on Information Theory . Subsystem code constructions . 0712.4321 . 2008 . 369–373. 10.1109/ISIT.2008.4595010 . 978-1-4244-2256-2 . 14063318 .
3. Fault-tolerant control of an error-corrected qubit. . Egan, L., Debroy, D.M., Noel, C. . Phys. Rev. Lett. . 2009.11482 . 7880 . 281–286 . 2021 . 598 . Nature . 10.1038/s41586-021-03928-y. 34608286 . 2021Natur.598..281E . 238357892 .
4. Stabilizer Formalism for Operator Quantum Error Correction . Poulin, David . Phys. Rev. Lett. . quant-ph/0508131 . 95 . 23 . 230504 . 2005 . American Physical Society . 10.1103/PhysRevLett.95.230504 . 16384287 . 2005PhRvL..95w0504P . 5348507 .
5. Subsystem fault tolerance with the Bacon-Shor code . Aliferis, Panos, Cross, Andrew W. . Phys. Rev. Lett. . quant-ph/0610063 . 98 . 22 . 220502 . 2007 . American Physical Society . 10.1103/PhysRevLett.98.220502 . 17677825 . 2007PhRvL..98v0502A . 11002341 .