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: [math]\displaystyle{ \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 }[/math], 4 stabilizers can be removed from this generator by recognizing gauge symmetries in the code to get: [math]\displaystyle{ \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 }[/math].[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:[math]\displaystyle{ \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 }[/math].[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
References
- ↑ Bacon, Dave (2006-01-30). "Operator quantum error-correcting subsystems for self-correcting quantum memories". Physical Review A 73 (1): 012340. doi:10.1103/PhysRevA.73.012340. Bibcode: 2006PhRvA..73a2340B.
- ↑ 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.
- ↑ 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.
- ↑ 4.0 4.1 Poulin, David (2005). "Stabilizer Formalism for Operator Quantum Error Correction". Phys. Rev. Lett. (American Physical Society) 95 (23): 230504. doi:10.1103/PhysRevLett.95.230504. PMID 16384287. Bibcode: 2005PhRvL..95w0504P. https://link.aps.org/doi/10.1103/PhysRevLett.95.230504.
- ↑ Aliferis, Panos, Cross, Andrew W. (2007). "Subsystem fault tolerance with the Bacon-Shor code". Phys. Rev. Lett. (American Physical Society) 98 (22): 220502. 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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