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By Eleanor Rieffel and Wolfgang Polak

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We have described Shor’s polynomialtime factorization algorithm, which stimulated the field of quantum computing. Given a practical quantum computer, Shor’s algorithm would make many present cryptographic methods obsolete. Grover’s search algorithm, while only providing a polynomial speed-up, proves that quantum computers are strictly more powerful than classical ones. Even though ACM Computing Surveys, Vol. 32, No. 3, September 2000. Introduction to Quantum Computing Grover’s algorithm has been shown to be optimal, there is hope that faster algorithms can be found by exploiting properties of the problem structure.

In any case, an error can be written as i ei Ei for some error operators Ei and coefficients ei . 2. Recovery of Quantum State An error correcting code for a set of errors Ei consists of a mapping C that embeds n data bits in n + k code bits together with a syndrome extraction operator SC that maps n + k code bits to the set of indices of correctable errors Ei such that i = SC (Ei (C(x))). , E S−1C ( y) ( y) = C(x)). Now consider the case of a quantum register. First, the state of the register can be in a superposition of basis vectors.

M. 1997. Secure communications using quantum cryptography. In S. P. HOTALING AND A. R. , Photonic Quantum Computing, Vol. 3076, 2–11. HUNGERFORD, T. A. 1974. Algebra. Springer Verlag, New York, Heidelberg, Berlin. JONES, J. A. AND MOSCA, M. 1998. Implementation of a quantum algorithm on a nuclear magnetic resonance quantum computer. Journal of Chemical Physics 109, 5, 1648–1653. gov/abs/quant-ph/9801027. , AND MARIAPPAN, S. 1997. NMR GHZ. Los Alamos ACM Computing Surveys, Vol. 32, No. 3, September 2000.

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An Introduction to Quantum Computing for Non-Physicists by Eleanor Rieffel and Wolfgang Polak

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