Bacon–Shor 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: $\langle X_{0}X_{1}X_{2}X_{3}X_{4}X_{5},X_{0}X_{1}X_{2}X_{6}X_{7}X_{8},Z_{0}Z_{1},Z_{1}Z_{2},Z_{3}Z_{4},Z_{4}Z_{5},Z_{6}Z_{7},Z_{7}Z_{8}\rangle$ , 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]

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

References

^ Bacon, Dave (2006-01-30). "Operator quantum error-correcting subsystems for self-correcting quantum memories". Physical Review A. 73 (1): 012340. arXiv:quant-ph/0506023. Bibcode:2006PhRvA..73a2340B. doi:10.1103/PhysRevA.73.012340. S2CID 118968017.
^ Aly Salah A., Klappenecker, Andreas (2008). "Subsystem code constructions". 2008 IEEE International Symposium on Information Theory. pp. 369–373. arXiv:0712.4321. doi:10.1109/ISIT.2008.4595010. ISBN 978-1-4244-2256-2. S2CID 14063318.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Egan, L., Debroy, D.M., Noel, C. (2021). "Fault-tolerant control of an error-corrected qubit". Phys. Rev. Lett. 598 (7880). Nature: 281–286. arXiv:2009.11482. Bibcode:2021Natur.598..281E. doi:10.1038/s41586-021-03928-y. PMID 34608286. S2CID 238357892.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ ^a ^b Poulin, David (2005). "Stabilizer Formalism for Operator Quantum Error Correction". Phys. Rev. Lett. 95 (23). American Physical Society: 230504. arXiv:quant-ph/0508131. Bibcode:2005PhRvL..95w0504P. doi:10.1103/PhysRevLett.95.230504. PMID 16384287. S2CID 5348507.
^ Aliferis, Panos, Cross, Andrew W. (2007). "Subsystem fault tolerance with the Bacon-Shor code". Phys. Rev. Lett. 98 (22). American Physical Society: 220502. arXiv:quant-ph/0610063. Bibcode:2007PhRvL..98v0502A. doi:10.1103/PhysRevLett.98.220502. PMID 17677825. S2CID 11002341.{{cite journal}}: CS1 maint: multiple names: authors list (link)

[1] Bacon, Dave (2006-01-30). "Operator quantum error-correcting subsystems for self-correcting quantum memories". Physical Review A. 73 (1): 012340. arXiv:quant-ph/0506023. Bibcode:2006PhRvA..73a2340B. doi:10.1103/PhysRevA.73.012340. S2CID 118968017.

[2] Aly Salah A., Klappenecker, Andreas (2008). "Subsystem code constructions". 2008 IEEE International Symposium on Information Theory. pp. 369–373. arXiv:0712.4321. doi:10.1109/ISIT.2008.4595010. ISBN 978-1-4244-2256-2. S2CID 14063318.{{cite book}}: CS1 maint: multiple names: authors list (link)

[3] Egan, L., Debroy, D.M., Noel, C. (2021). "Fault-tolerant control of an error-corrected qubit". Phys. Rev. Lett. 598 (7880). Nature: 281–286. arXiv:2009.11482. Bibcode:2021Natur.598..281E. doi:10.1038/s41586-021-03928-y. PMID 34608286. S2CID 238357892.{{cite journal}}: CS1 maint: multiple names: authors list (link)

[poulin-4] Poulin, David (2005). "Stabilizer Formalism for Operator Quantum Error Correction". Phys. Rev. Lett. 95 (23). American Physical Society: 230504. arXiv:quant-ph/0508131. Bibcode:2005PhRvL..95w0504P. doi:10.1103/PhysRevLett.95.230504. PMID 16384287. S2CID 5348507.

[5] Aliferis, Panos, Cross, Andrew W. (2007). "Subsystem fault tolerance with the Bacon-Shor code". Phys. Rev. Lett. 98 (22). American Physical Society: 220502. arXiv:quant-ph/0610063. Bibcode:2007PhRvL..98v0502A. doi:10.1103/PhysRevLett.98.220502. PMID 17677825. S2CID 11002341.{{cite journal}}: CS1 maint: multiple names: authors list (link)

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Bacon–Shor code

See also

References