Bacon–Shor code

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Short description: Quantum error correcting code

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: ⟨X0X1X2X3X4X5,X0X1X2X6X7X8,Z0Z1,Z1Z2,Z3Z4,Z4Z5,Z6Z7,Z7Z8⟩, 4 stabilizers can be removed from this generator by recognizing gauge symmetries in the code to get: ⟨X0X1X2X3X4X5,X0X1X2X6X7X8,Z0Z1Z3Z4Z6Z7,Z1Z2Z4Z5Z7Z8⟩.[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:⟨Z1Z2,X2X8,Z4Z5,X5X8,Z0Z1,X0X6,Z3Z4,X3X6,X1X7,X4X7,Z6Z7,Z7Z8⟩.[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 overhead for fault tolerant error correction.[5]

Geometry
q0 ZZ q1 ZZ q2
XX XX XX
q6 ZZ q7 ZZ q8
XX XX XX
q3 ZZ q4 ZZ q5


See also

References

  1. ↑ Bacon, Dave (2006-01-30). "Operator quantum error-correcting subsystems for self-correcting quantum memories". Physical Review A 73 (1). doi:10.1103/PhysRevA.73.012340. Bibcode: 2006PhRvA..73a2340B. 
  2. ↑ Aly Salah A., Klappenecker, Andreas (2008). "Subsystem code constructions". 2008 IEEE International Symposium on Information Theory. pp. 369–373. doi:10.1109/ISIT.2008.4595010. ISBN 978-1-4244-2256-2. 
  3. ↑ Egan, L., Debroy, D.M., Noel, C. (2021). "Fault-tolerant control of an error-corrected qubit.". Phys. Rev. Lett. (Nature) 598 (7880): 281–286. doi:10.1038/s41586-021-03928-y. PMID 34608286. Bibcode: 2021Natur.598..281E. 
  4. ↑ 4.0 4.1 Poulin, David (2005). "Stabilizer Formalism for Operator Quantum Error Correction". Phys. Rev. Lett. (American Physical Society) 95 (23). doi:10.1103/PhysRevLett.95.230504. PMID 16384287. Bibcode: 2005PhRvL..95w0504P. https://link.aps.org/doi/10.1103/PhysRevLett.95.230504. 
  5. ↑ Aliferis, Panos, Cross, Andrew W. (2007). "Subsystem fault tolerance with the Bacon-Shor code". Phys. Rev. Lett. (American Physical Society) 98 (22). doi:10.1103/PhysRevLett.98.220502. PMID 17677825. Bibcode: 2007PhRvL..98v0502A. https://link.aps.org/doi/10.1103/PhysRevLett.98.220502. 



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