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.

翻译：我们考虑在噪声量子门设定下的肖尔量子质因数分解算法。在（受控）旋转门的随机噪声通用模型下，我们证明：当噪声超出关于待分解整数比特数 $n$ 的极低水平时，该算法无法分解形如 $pq$ 的整数，其中 $p$ 和 $q$ 属于特定正密度素数集合。我们进一步证明：在存在相同随机噪声水平的情况下，对于随机素数对 $(p,q)$，肖尔分解算法以 $1 - o(1)$ 的概率无法分解形如 $pq$ 的数。