Editing Quantum error correction (section)

==Models==

Over time, researchers have come up with several codes:

* [[Peter Shor]]'s 9-qubit-code, a.k.a. the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for arbitrary errors in a single qubit.
* [[Andrew Steane]] found a code that does the same with 7 instead of 9 qubits, see [[Steane code]].
* [[Raymond Laflamme]] and collaborators found a class of 5-qubit codes that do the same, which also have the property of being [[fault-tolerant]]. A [[Five-qubit error correcting code|5-qubit code]] is the smallest possible code that protects a single logical qubit against single-qubit errors.
* A generalisation of the technique used by [[Andrew Steane|Steane]], to develop the [[Steane code|7-qubit code]] from the [[Hamming(7,4)|classical [7, 4] Hamming code]], led to the construction of an important class of codes called the [[CSS code]]s, named for their inventors: [[Robert Calderbank]], Peter Shor and Andrew Steane. According to the quantum Hamming bound, encoding a single logical qubit and providing for arbitrary error correction in a single qubit requires a minimum of 5 physical qubits.
* A more general class of codes (encompassing the former) are the [[stabilizer code]]s discovered by [[Daniel Gottesman]], and by [[Robert Calderbank]], [[Eric Rains]], Peter Shor, and [[N. J. A. Sloane]]; these are also called [[additive code]]s.
*Two dimensional [[Bacon–Shor code]]s are a family of codes parameterized by integers ''m'' and ''n''. There are ''nm'' qubits arranged in a square lattice.<ref>{{Cite journal|last=Bacon|first=Dave|date=2006-01-30|title=Operator quantum error-correcting subsystems for self-correcting quantum memories|journal=Physical Review A|volume=73|issue=1| pages=012340| doi=10.1103/PhysRevA.73.012340| arxiv=quant-ph/0506023|bibcode=2006PhRvA..73a2340B| s2cid=118968017}}</ref> 
* [[Alexei Kitaev]]'s [[toric code|topological quantum code]]s, introduced in 1997 as the toric code, and the more general idea of a [[topological quantum computer]] are the basis for various code types.<ref>
{{ cite conference
|url= https://link.springer.com/book/10.1007/978-1-4615-5923-8
|last= Kitaev
|first= Alexei
|title= Quantum Error Correction with Imperfect Gates
|date= July 31, 1997
|publisher= Springer 
|doi= 10.1007/978-1-4615-5923-8
|book-title= Quantum Communication, Computing, and Measurement
|pages= 181–188
}}
</ref>
* [[Todd Brun]], [[Igor Devetak]], and [[Min-Hsiu Hsieh]] also constructed the [[entanglement-assisted stabilizer formalism]] as an extension of the standard [[stabilizer formalism]] that incorporates [[quantum entanglement]] shared between a sender and a receiver.
* The ideas of stabilizer codes, CSS codes, and topological codes can be expanded into the 2D planar [[surface code]], of which various types exist.<ref>{{cite journal | last1=Fowler | first1=Austin G. | last2=Mariantoni | first2=Matteo | last3=Martinis | first3=John M. | last4=Cleland | first4=Andrew N. | title=Surface codes: Towards practical large-scale quantum computation | journal=Physical Review A | volume=86 | issue=3 | date=2012-09-18 | page=032324 | issn=1050-2947 | doi=10.1103/PhysRevA.86.032324| arxiv=1208.0928 | bibcode=2012PhRvA..86c2324F }}</ref> As of June 2024, the 2D planar surface code is generally considered the most well-studied type of quantum error correction, and one of the leading contenders for practical use in quantum computing.<ref>{{cite arXiv | last1=Gidney | first1=Craig | last2=Newman | first2=Michael | last3=Brooks | first3=Peter | last4=Jones | first4=Cody | title=Yoked surface codes | date=2023 | class=quant-ph | eprint=2312.04522 }}</ref><ref>{{cite journal | last1=Horsman | first1=Dominic | last2=Fowler | first2=Austin G | last3=Devitt | first3=Simon | last4=Meter | first4=Rodney Van | title=Surface code quantum computing by lattice surgery | journal=New Journal of Physics | volume=14 | issue=12 | date=2012-12-01 | issn=1367-2630 | doi=10.1088/1367-2630/14/12/123011 | page=123011| arxiv=1111.4022 | bibcode=2012NJPh...14l3011H }}</ref>

That these codes allow indeed for quantum computations of arbitrary length is the content of the [[quantum threshold theorem]], found by [[Michael Ben-Or]] and [[Dorit Aharonov]], which asserts that you can correct for all errors if you concatenate quantum codes such as the CSS codes—i.e. re-encode each logical qubit by the same code again, and so on, on logarithmically many levels—''provided'' that the error rate of individual [[quantum gate]]s is below a certain threshold; as otherwise, the attempts to measure the syndrome and correct the errors would introduce more new errors than they correct for.

As of late 2004, estimates for this threshold indicate that it could be as high as 1–3%,<ref>{{cite journal
|last= Knill
|first= Emanuel
|arxiv= quant-ph/0410199
|title= Quantum Computing with Very Noisy Devices
|date= November 2, 2004
|doi=10.1038/nature03350
|pmid= 15744292
|volume=434
|issue= 7029
|journal=Nature
|pages=39–44
|bibcode=2005Natur.434...39K
|s2cid= 4420858
}}</ref> provided that there are sufficiently many [[qubit]]s available.