Shor’s algorithm does not factor large integers in the presence of noise

We consider Shor’s quantum factoring algorithm in the setting of noisy quantum gates. Under a generic model of random noise for (controlled) rotation gates, we prove that the algorithm does not factor integers of the form pq when the noise exceeds a vanishingly small level in terms of n—the number of bits of the integer to be factored, where p and q are from a well-defined set of primes of positive density. We further prove that with probability 1 − o(1) over random prime pairs (p, q), Shor’s factoring algorithm does not factor numbers of the form pq, with the same level of random noise present.