We heuristically show that Shor's algorithm for computing general discrete logarithms achieves an expected success probability of approximately 60% to 82% in a single run when modified to enable efficient implementation with the semi-classical Fourier transform. By slightly increasing the number of group operations that are evaluated quantumly and performing a single limited search in the classical post-processing, or by performing two limited searches in the post-processing, we show how the algorithm can be further modified to achieve a success probability that heuristically exceeds 99% in a single run. We provide concrete heuristic estimates of the success probability of the modified algorithm, as a function of the group order $r$, the size of the search space in the classical post-processing, and the additional number of group operations evaluated quantumly. In the limit as $r \rightarrow \infty$, we heuristically show that the success probability tends to one. In analogy with our earlier works, we show how the modified quantum algorithm may be heuristically simulated classically when the logarithm $d$ and $r$ are both known. Furthermore, we heuristically show how slightly better tradeoffs may be achieved, compared to our earlier works, if $r$ is known when computing $d$. We generalize our heuristic to cover some of our earlier works, and compare it to the non-heuristic analyses in those works.

翻译：我们启发式地表明，当修改Shor算法以支持通过半经典傅里叶变换实现高效实现时，该算法在单次运行中能达到约60%至82%的预期成功概率。通过略微增加量子计算评估的群运算数量并在经典后处理中执行一次有限搜索，或在后处理中执行两次有限搜索，我们展示了如何进一步修改算法以在单次运行中实现启发式超过99%的成功概率。我们提供了修改后算法成功概率的具体启发式估计，该估计是群阶$r$、经典后处理中搜索空间大小以及量子计算评估的额外群运算数量的函数。当$r \rightarrow \infty$时，我们启发式地证明成功概率趋近于一。类比我们早期的研究工作，我们展示了当对数$d$和$r$均已知时，如何启发式地经典模拟修改后的量子算法。此外，我们启发式地展示了在计算$d$时若已知$r$，相比我们早期工作可能实现略优的权衡。我们将启发式方法推广至涵盖部分早期工作，并将其与那些工作中的非启发式分析进行比较。