We establish a vanishingly small noise threshold for Shor's factoring and discrete log algorithms. Below ε/2^b = O(log n/n^1/2), algorithms succeed in polynomial time; above it, they provably fail for positive-density prime sets.

We establish a vanishingly small noise threshold for Shor's factoring and discrete log algorithms. Below ε/2^b = O(log n/n^1/2), algorithms succeed in polynomial time; above it, they provably fail for positive-density prime sets.