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Quantum error correction is used in Quantum Computing to protect Quantum Information from errors due to Decoherence and other quantum noise Noise . Quantum error correction is a key building block for the fault-tolerant quantum computation which is designed to deal not just with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation procedures, and faulty measurements. INTRODUCTION Classical Error Correction employs Redundancy : The simplest way is to store the information multiple times, and—if these copies are later found to disagree—just take a majority vote; i.e. if, say, one copy says, the bit is a 0, and two others claim it to be a 1, then probably all three were a 1 and the first bit got corrupted. Although copying is not possible with quantum information due to No-cloning Theorem , the information of one Qubit may be ''spread'' onto several (''physical'') qubits by using a ''quantum error correcting code''. Such ''encoded'' quantum information is protected, as in classical error correcting codes, against errors of a limited form. As in classical error correcting codes, a ''syndrome measurement'' can determine whether a qubit has been corrupted, and if so, which one. What is more, the outcome of this operation (the ''syndrome'') tells us not only which physical qubit was affected, but also, in which of several possible ways it was affected. The latter is counter-intuitive at first sight: Since noise is arbitrary, how can the effect of noise be one of only few distinct possibilities? In most codes, the effect is either a bit flip, or a sign (of the Phase ) flip, or both (corresponding to the Pauli Matrices ''X'', ''Z'', and ''Y''). The reason is that the measurement of the syndrome has the Projective effect of a Quantum Measurement . So even if the error due to the noise was arbitrary, it can be expressed as a Superposition of Basis operations—the ''error basis'' (which is here given by the Pauli matrices and the Identity ). The syndrome measurement "forces" the qubit to "decide" for a certain specific "Pauli error" to "have happened", and the syndrome tells us which, so that we can let the same Pauli operator act again on the corrupted qubit to revert the effect of the error. The crucial point is that the syndrome measurement tells us as much as possible about the error that has happened, but ''nothing'' at all about the ''value'' that is stored in the logical qubit—as otherwise the measurement would destroy any Quantum Superposition of this logical qubit with other qubits in the Quantum Computer . Over time, researchers have come up with several codes:
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 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. Recent (as of late 2004) estimates for this threshold indicate that it could be as high as 1-3% {Link without Title} , provided that there are sufficiently many Qubits available. EXTERNAL LINKS |
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