On the Complexity of Shor´s Algorithm for Factorization

The complexity analysis of Shor´s quantum algorithm for factorization consists of: 1) The probability p that we see any particular statec, x^k (mod n)) with {rc}_q ¿ r/2 is at least 1/3r². 2) There are ¿(r) possible values of c, and r possible values of x^k (mod n). 3) The success probability is at least ¿(r) · r · 1/3r². That is, the inventor views p as the joint probability P(X = c, Y = x^k (mod n)). In this paper, we show that the argument for the estimation of P(X = c, Y = x^k (mod n)) is not sound. Therefore, the problem that Shor´s algorithm takes polynomial time remains open.