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• CommentTimeFeb 7th 2020

stub

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• CommentTimeFeb 7th 2020

still stub

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• CommentTimeFeb 7th 2020

• Alexander Jahn, Marek Gluza, Fernando Pastawski, Jens Eisert, Majorana dimers and holographic quantum error-correcting codes, Phys. Rev. Research 1, 033079 (2019) (arXiv:1905.03268)
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 11th 2020

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• CommentTimeFeb 14th 2020

• Simon J. Devitt, Kae Nemoto, William J. Munro, Quantum Error Correction for Beginners, Rep. Prog. Phys. 76 (2013) 076001 (arXiv:0905.2794)
• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeFeb 12th 2021

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 20th 2021

added these references on operator algebraic formulation of quantum error correction in the Heisenberg picture:

• Cédric Bény, Achim Kempf, David W. Kribs, Generalization of Quantum Error Correction via the Heisenberg Picture, Phys. Rev. Lett. 98, 100502 – Published 7 March 2007 (doi:10.1103/PhysRevLett.98.100502, arXiv:quant-ph/0608071)

• Cédric Bény, Achim Kempf, David W. Kribs, Quantum Error Correction of Observables, Phys. Rev. A 76, 042303 (2007) (arXiv:0705.1574)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMay 2nd 2021

I have tried my hand on some first linea of an Idea-section, and then added the simple explicit example (here) of a qtrit error correcting code from CGL 99 as highlighted in ADH 14.

Still need to add a more abstract definition/characterization…

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMay 2nd 2021
• (edited May 2nd 2021)

• Andrew J. Ferris, David Poulin, Tensor Networks and Quantum Error Correction, Phys. Rev. Lett. 113, 030501 (2014) (arXiv:1312.4578)

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• CommentTime5 days ago

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• CommentTime5 days ago

• Sam Cree, Kfir Dolev, Vladimir Calvera, Dominic J. Williamson, Fault-tolerant logical gates in holographic stabilizer codes are severely restricted (arXiv:2103.13404)
• CommentRowNumber13.
• CommentAuthorUrs
• CommentTime5 days ago

that last article (CDCW 21) comments on the perspective of practical usefulness of holographic codes (finally, I kept looking for this in vain).

I have added the following quote from the article to the Idea-section of the entry:

There are a number of reasons to suspect that holographic codes may be of practical use for quantum computing.

Holographic codes can admit erasure thresholds comparable to that of the widely-studied surfacecode, and likewise for their threshold against Pauli errors. Their holographic structure also naturally leads to an organization of encoded qubits into a hierarchy of levels of protection from errors, which could be useful for applications which call for many qubits withvarying levels of protection. In particular, this is reminiscent of many schemes for magic state distillation – and indeed, the concatenated codes utilized for magic state distillation share a similar hierarchical structure to holographic codes. The layered structure of holographic codes is also reminiscent of memory architectures in classical computers, where it is useful to have different levels of short- and long-term memory. Although these codes have some notable drawbacks, in particular holographic stabilizer codes require nonlocal stabilizer generators, other codes such as concatenated codes suffer similar drawbacks and have still proven to be useful. Conversely, the stringent requirement of non-local stabilizer generators allows holographic codes to protect many more qubits than a topological code and in fact attain a finite nonzero encoding rate, which is typically not possible for topological codes. Nonetheless, many open questions remain about the usefulness of holographic codes for fault-tolerant quantum computing.

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• CommentTime5 days ago

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• CommentRowNumber16.
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• CommentTime5 days ago
• (edited 5 days ago)

$[$ holographic codes $]$ could be promising candidates to circumvent our results and could possibly realise a universal set of unitary implementations of logical operators.