Operator-algebra error-correcting code

Operator-algebra error-correcting code[1–6]

Stub.

Error-correcting code (ECC) — Any ECC can be embedded into a quantum Hilbert space, and thus passed through a quantum channel, by associating elements of the alphabet with basis vectors in a Hilbert space over the complex numbers. For example, a bit of information can be embedded into a two-dimensional vector space by associating the two bit values with two basis vectors for the space.
Classical-quantum (c-q) code
Entanglement-assisted hybrid classical-quantum (EACQ) code — EACQ codes utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to operator-algebra QECCs when said subsystems are interpreted as noiseless physical subsystems.
Subsystem quantum error-correcting code
Quantum error-correcting code (QECC)

Holographic code — Properties of holographic codes are often quantified in the Heisenberg picture, i.e., in terms of operator algebras [7,8].

[1]: G. Kuperberg, “The capacity of hybrid quantum memory”, (2003) arXiv:quant-ph/0203105
[2]: C. Bény, A. Kempf, and D. W. Kribs, “Generalization of Quantum Error Correction via the Heisenberg Picture”, Physical Review Letters 98, (2007) arXiv:quant-ph/0608071 DOI
[3]: C. Bény, A. Kempf, and D. W. Kribs, “Quantum error correction of observables”, Physical Review A 76, (2007) arXiv:0705.1574 DOI
[4]: G. Kuperberg and N. Weaver, “A von Neumann algebra approach to quantum metrics”, (2010) arXiv:1005.0353
[5]: C. BÉNY, D. W. KRIBS, and A. PASIEKA, “ALGEBRAIC FORMULATION OF QUANTUM ERROR CORRECTION”, International Journal of Quantum Information 06, 597 (2008) DOI
[6]: K. Furuya, N. Lashkari, and S. Ouseph, “Real-space RG, error correction and Petz map”, Journal of High Energy Physics 2022, (2022) arXiv:2012.14001 DOI
[7]: A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT”, Journal of High Energy Physics 2015, (2015) arXiv:1411.7041 DOI
[8]: F. Pastawski and J. Preskill, “Code Properties from Holographic Geometries”, Physical Review X 7, (2017) arXiv:1612.00017 DOI

Page edit log

“Operator-algebra error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/oaecc