Realistic lower bounds for the factorization time of large numbers on a quantum computer.

AbstractWe investigate the time T a quantum computer requires to factorize a givennumber dependent on the number of bits L required to represent this number. Westressthefactthatinmostcasesonehastotakeintoaccount thattheexecution timeof a single quantum gate is related to the decoherence time of the qubits that areinvolved in the computation. Although exhibited here only for special systems, thisinter-dependence of decoherence and computation time seems to be a restrictionin many current models for quantum computers and leads to the result that thecomputation time T scales much stronger with L than previously expected.PACS: 42.50.Lc I. IntroductionSince Shor’s discovery [1, 2] of an algorithm that allows thefactorization of a large numberby a quantum computer in polynomial time instead of an exponential time as in classicalcomputing, interest in the practical realization of a quantum computer has been muchenhanced. Recent advances in the preparation and manipulation of single ions as well asthe engineering of pre-selected cavity light fields have made quantum optics that field ofphysics which promises the first experimental realization of a quantum computer. Severalproposals for possible experimental implementations have been made relying on nuclearspins, quantum dots [3], cavity QED [4] and on ions in linear traps [5].One can estimate the time T needed for a single run of Shor’s algorithm to be equalto the time τ