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Bacon–Shor code

Today, Bacon–Shor code remains a topic of constant interest and debate in modern society. With increasing interest in recent years, Bacon–Shor code has captured the attention of experts and fans alike. Whether in academia, in the media, or in everyday conversations, Bacon–Shor code has become a central point of discussion. This trend is not only on a local level, but has also gained relevance on a global level, demonstrating the importance and impact that Bacon–Shor code has on our current reality. Given this growing attention, it is crucial to thoroughly analyze the various dimensions and perspectives that Bacon–Shor code involves, to better understand its scope and implications in our society.

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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: , 4 stabilizers can be removed from this generator by recognizing gauge symmetries in the code to get: .[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:.[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) 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)
  4. ^ a b Poulin, David (2005). "Stabilizer Formalism for Operator Quantum Error Correction". Phys. Rev. Lett. 95 (23) 230504. American Physical Society. 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) 220502. American Physical Society. 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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