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 which does the same with 7 instead of 9 qubits, see Steane code.
- Raymond Laflamme found a class of 5-qubit codes which do the same, which also have the property of being fault-tolerant.
- A generalisation of this concept are the CSS codes, named for their inventors: A. R. 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 codes discovered by Daniel Gottesman, and by A. R. Calderbank, Eric Rains, Peter Shor, and N. J. A. Sloane (, ); these are also called additive codes.
- A newer idea is Alexei Kitaev's topological quantum codes and the more general idea of a topological quantum computer.
- 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.
That these codes allow indeed for quantum computations of arbitrary length is the content of the 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 the error rate of individual quantum gates 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%, provided that there are sufficiently many qubits available.
Read more about this topic: Quantum Error Correction
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