Shor's Algorithm Does Not Factor Large Integers in the Presence of Noise
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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.
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