#First quantum error correcting code code

On the Significance of the Gottesman-Knill Theorem, Michael E. On the role of entanglement in quantum computational speed-up, Jozsa and Linden,,

Quantum Computation and Quantum Information: 10th Anniversary Edition, Nielsen, and I Chuang.

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.Д¬onsider the situation in which we want to transmit the state of a single qubit | П€ вџ© 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. 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 reason is that the measurement of the syndrome has the projective effect of a quantum measurement. 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). 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. A syndrome measurement can determine whether a qubit has been corrupted, and if so, which one. We perform a multi-qubit measurement that does not disturb the quantum information in the encoded state but retrieves information about the error. Quantum error correction also employs syndrome measurements. We then reverse an error by applying a corrective operation based on the syndrome.

#First quantum error correcting code code

A quantum error correcting code protects quantum information against errors of a limited form.Д¬lassical error correcting codes use a syndrome measurement to diagnose which error corrupts an encoded state. Peter Shor first discovered this method of formulating a quantum error correcting code by storing the information of one qubit onto a highly entangled state of nine qubits. But it is possible to spread the information of one qubit onto a highly entangled state of several ( physical) qubits. This theorem seems to present an obstacle to formulating a theory of quantum error correction. It is possible that a double-bit error occurs and the transmitted message is equal to three zeros, but this outcome is less likely than the above outcome.Д¬opying quantum information is not possible due to the no-cloning theorem. If we assume that noisy errors are independent and occur with some probability p, it is most likely that the error is a single-bit error and the transmitted message is three ones. Suppose further that a noisy error corrupts the three-bit state so that one bit is equal to zero but the other two are equal to one. The simplest way is to store the information multiple times, and-if these copies are later found to disagree-just take a majority vote e.g. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.Д¬lassical error correction employs redundancy. Quantum error correction ( QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise.