Science, 8 December 1995 Quantum Computers, Factoring, and Decoherence I. L. Chuang, R. Laflamme, P. W. Shor, W. H. Zurek [First paragraph] The uniqueness of the prime factorization of a positive integer is the Fundamental Theorem of Arithmetic. In practice, the determination of the prime factors of a given number can be an exceedingly difficult problem, even though verification is trivial. This asymmetry is the basis for modern cryptography and provides secret codes used not only on your own bank card but also to transfer diplomatic messages between embassies. [Precis] It is known that quantum computers can dramatically speed up the task of finding factors of large numbers, a problem of practical significance for cryptographic applications. Factors of an L-digit number can be found in ~L^2 time [compared to ~exp(L^1/3) time] by a quantum computer, which simultaneously follows all paths corresponding to distinct classical inputs, obtaining the solution from the coherent quantum interference of the alternatives. Here it is shown how the decoherence process degrades the interference pattern that emerges from the quantum factoring algorithm. For a quantum computer performing logical operations, an exponential decay of quantum coherence is inevitable. However, even in the presence of exponential decoherence, quantum computation can be useful as long as a sufficiently low decoherence rate can be achieved to allow meaningful results to be extracted from the calculation. I. L. Chuang, Stanford University. R. Laflamme and W. H. Zurek, Los Alamos National Laboratory. P. W. Shor, AT&T Bell Labs. ---------- QCF_dec (18 kb) Sent with compressed qcf1.jpg of 14 equations and 2 figures (31 kb)