Polynomial time algorithms for discrete logarithms and factoring on a quantum computer

Abstract

A computer is generally considered to be a universal computational device, in other words, it is able to simulate any physical computational device with a cost of at most a polynomial factor in the computation time. It is not clear that this is still true if quantum mechanics is taken into account. Feynman seems to have been the first to ask what effect the non-local properties of quantum mechanics have on computation [7, 8]. He gave arguments as to why these properties might make it intrinsically computationally expensive to simulate quantum mechanics on a classical (Von Neumann) computer. He also asked the converse question: whether these properties permit more powerful computation. Several researchers have since developed models for quantum mechanical computers and investigated their computational properties. [4, 5, 6, 2, 3, 1, 10, 9] We give Las Vegas algorithms for the discrete logarithm and integer factoring problems that take random polynomial time on a quantum computer.