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.
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A quantum error correcting code protects quantum information against errors of a limited form.Ĭlassical 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.Ĭopying quantum information is not possible due to the no-cloning theorem. We also 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.Ĭlassical error correction employs redundancy. I am not an expert on 6 or 7, but doing a search on for those topics should throw up some interesting papers.Quantum error correction is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. This will give you a good picture of the field up to about 1997.Īlso recommended is his introductory article. Of course, you probably only want to cover two or three of these topics.Ī good starting point is Dan Gottesman's Ph.D.
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collective decoherence and encoding in decoherence free subspaces. Nonstandard methods of error correction, e.g. Experimental implementations of error correction codes (these have mainly been done in NMR).ħ. Fault tolerance - how error correction can be used to prove the threshold theorem for quantum computation.Ħ.
Gottesman's stabilizer formalism for quantum error correcting codes and its connection to classical error correction codes.ĥ. Simple examples of error correction codes - quantum repetition code, Shor's nine-qubit code, CSS codes and the 5 qubit code.Ĥ. The ``standard model'' of quantum error correction and how it leads to the requirements and definitions of quantum error correction codes.Ģ. As I see it, the main topics you could choose to cover are:ġ. I have also written a dissertation on quantum error correction, although that was in graduate school, so maybe mine was more technical than yours has to be.
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