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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)
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and added a quote from this article to the Idea-section
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John Preskill, Reliable Quantum Computers, Proc. Roy. Soc. Lond. A454 (1998) 385-410 (arXiv:quant-ph/9705031)
Dorit Aharonov, Michael Ben-Or, Fault-Tolerant Quantum Computation With Constant Error Rate, SIAM J. Comput., 38(4), 1207–1282. (arXiv:quant-ph/9906129, doi:10.1137/S0097539799359385)
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S. B. Bravyi, Alexei Kitaev, Quantum codes on a lattice with boundary (arXiv:quant-ph/9811052)
Michael Freedman, David A. Meyer, Projective Plane and Planar Quantum Codes, Found. Comput. Math. 1, 325–332 (2001) (arXiv:quant-ph/9810055, doi:10.1007/s102080010013)
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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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with this quote:
$[$ holographic codes $]$ could be promising candidates to circumvent our results and could possibly realise a universal set of unitary implementations of logical operators.
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with this quote:
Although we are currently in an era of quantum computers with tens of noisy qubits, it is likely that a decisive, practical quantum advantage can only be achieved with a scalable, fault-tolerant, error-corrected quantum computer. Therefore, development of quantum error correction is one of the central themes of the next five to ten years.
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Daniel Nigg, Markus Mueller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel A. Martin-Delgado, Rainer Blatt,
Experimental Quantum Computations on a Topologically Encoded Qubit, Science 18 Jul 2014: Vol. 345, Issue 6194, pp. 302-305 (arXiv:1403.5426, doi:10.1126/science.1253742)
and am also adding this to topological quantum computation with anyons – references
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with this quote:
The codes provided by AdS/CFT often come close to saturating theoretical bounds on the performance of quantum codes. It seems AdS/CFT may be a tool for discovering better quantumcryptography?
Even though it’s promotional, I have added this pointer, from just days ago
with the following quote, since it’s the best reference I can find, so far, on outlook/expectation on the pracrtical realizability of quantum error correction:
Within the decade, Google aims to build a useful, error-corrected quantum computer. $[\cdots]$ Our journey to build an error-corrected quantum computer within the decade includes several scientific milestones, including building an error-corrected logical qubit.
on the other hand, over in China:
Ming Gong et al. Experimental exploration of five-qubit quantum error correcting code with superconducting qubits, National Science Review, nwab011 (2021) (doi:10.1093/nsr/nwab011)
survey in:
EurekaAlert, Science China Press: Demonstration of the universal quantum error correcting code with superconducting qubits, March 2021 (2021-03/scp-dot031521)
The 21st century version of the Space Race.
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