= Quantum error correction
{title2=QEC}
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Quantum error correction protects a <logical qubit> by encoding its <quantum state> into a larger <Hilbert space>, where a specified family of physical errors can be diagnosed and reversed by a common recovery <quantum channel>. The requirement is to restore every encoded <quantum superposition>, including its relative phase, rather than only the <computational basis> states. A <quantum error-correcting code> specifies which errors have this protection; it does not promise recovery from every possible disturbance.
The beautiful idea is to measure relations between physical <qubits> that reveal an <error syndrome> while revealing nothing about the protected <logical qubit>. For example, the <bit-flip repetition code> compares neighboring <qubits> without reading their individual values. Its parity checks can locate one <Pauli X gate> error while leaving the unknown coefficients intact. Encoding is a <linear isometry>, consistent with the <no-cloning theorem>: redundancy in a shared <quantum state> supplies protection without making independent copies of the unknown input.
Peter W. Shor's 1995 paper https://journals.aps.org/pra/abstract/10.1103/PhysRevA.52.R2493[Scheme for reducing decoherence in quantum computer memory]{ref}, published on 1 October, introduced the nine-<qubit> <Shor code> to reduce <quantum decoherence> in memory. Andrew M. Steane's https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.77.793[Error Correcting Codes in Quantum Theory]{ref}, published on 29 July 1996 and https://users.physics.ox.ac.uk/~Steane/pubs/Steane_PRL95.pdf[available from its author's site]{ref}, connected <quantum error correction> to classical <linear codes>; the seven-<qubit> <Steane code> protects against arbitrary errors on one physical <qubit>. These constructions protect relative phase as well as <computational basis> values, going beyond the elementary <bit-flip repetition code>.
For an authored account of the mechanism, see John Preskill's https://www.preskill.caltech.edu/ph229/notes/chap7.pdf[Quantum Error Correction lecture notes]{ref}, especially section 7.1 on collective parity measurements and section 7.9 on <stabilizer codes>. Andrew Steane's https://users.physics.ox.ac.uk/~Steane/pubs/Steane_2006.pdf[A tutorial on quantum error correction] develops the distinction between correcting a stored <quantum state> and operating with imperfect <quantum gates>.
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