lect notes on quantum computation(lect5)

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Last revised 5/3/06 LECTURE NOTES ON QUANTUM COMPUTATION Cornell University, Physics 481-681, CS 483; Spring, 2006 V. Quantum Error Correction. c 2006, N. David Mermin Correcting errors might sound like a dreary practical problem, of little aesthetic or conceptual interest. But aside from being of crucial importance for the feasibility of quan- tum computation, it is also one of the most beautiful and surprising parts of the subject. The surprise is that error correction is possible at all, since the only way to detect er- rors is to make measurements. But measurement gates disruptively alter the states of the measured Qbits, apparently making things even worse. “Quantum error correction” would seem to be an oxymoron. The beauty, which I hope will emerge below, lies in the ele- gant and ingenious ways that people have found to get around this apparently insuperable obstacle. The discovery in 1995 of quantum error correction by Peter Shor and, independently, Andrew Steane, had an enormous impact on the prospects for actual quantum computa- tion. It changed the dream of building a quantum computer, capable of useful computation, from a clearly unattainable vision, to a program that poses an enormous but not necessarily insuperable technological challenge. Error correction is simply not an issue in classical computation. In a classical computer the physical systems that embody individual bits — the Cbits — are immense on the atomic scale. The two states of a Cbit representing 0 and 1 are so grossly different that the probability is infinitesimal for flipping from one to the other as a result of thermal fluctuations, mechanical vibrations, or other irrelevant extraneous interactions. Error correction does become an issue, even classically, in the transmission of infor- mation over large distances, because the farther the signal travels, the more it attenuates. One can deal with this in a variety of straightforward or ingenious ways. One of the crudest is to encode each logical bit in three actual bits, replacing | 0 i and | 1 i with the codewords | 0 i = | 0 i| 0 i| 0 i = | 000 i , | 1 i = | 1 i| 1 i| 1 i = | 111 i . (5 . 1) One can then monitor each codeword, checking for flips in any of the individual Cbits and restoring them by the principle of majority rule, whenever a flip is detected. Monitoring has to take place often enough to make negligible the probability that more than a single bit flips in a single codeword between inspections. Quantum error correction also uses multi-Qbit codewords and it requires monitoring at a rate that renders certain kinds of errors highly improbable. But there are several ways in which error correction in a quantum computer is quite different: 1
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(a) In a quantum computer, unlike a classical computer, error correction is essential.
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