We prove a lower bound on the probability of Shor's order-finding algorithm successfully recovering the order $r$ in a single run. The bound implies that by performing two limited searches in the classical post-processing part of the algorithm, a high success probability can be guaranteed, for any $r$, without re-running the quantum part or increasing the exponent length compared to Shor. Asymptotically, in the limit as $r$ tends to infinity, the probability of successfully recovering $r$ in a single run tends to one. Already for moderate $r$, a high success probability exceeding e.g. $1 - 10^{-4}$ can be guaranteed. As corollaries, we prove analogous results for the probability of completely factoring any integer $N$ in a single run of the order-finding algorithm.

翻译：我们证明Shor的订单调查算法在一次运行中成功收回订单的概率上存在较低的约束。 约束意味着通过在典型的后处理算法部分进行两次有限的搜索,可以保证任何美元都具有很高的成功概率,而无需重新运行量部分,或增加与Shor相比的超长。 顺理成章,由于美元往往具有无限性,因此在一次运行中成功收回1美元的可能性就只有1美元。 对于中等的美元来说,成功概率可能超过1美元-10美元-4美元就可以得到保证。 作为卷轴,我们证明,在单次运行的订单调查算法中,将任何整数一美元完全乘以任何整数的概率是相似的结果。