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：我们启发式地证明，在采用半经典傅里叶变换进行高效实现时，Shor算法计算一般离散对数在单次运行中的期望成功概率约为60%至82%。通过适度增加量子计算的群运算次数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，我们进一步改进算法，使其单次运行的成功概率启发式地超过99%。我们给出了改进算法成功概率的具体启发式估计，该估计依赖于群阶$r$、经典后处理中的搜索空间大小以及量子计算的额外群运算次数。在$r \rightarrow \infty$的极限情况下，我们启发式地证明成功概率趋近于1。与早期工作类似，我们展示了当对数$d$和$r$均已知时，如何用经典方法启发式地模拟改进后的量子算法。此外，我们启发式地证明，当计算$d$时已知$r$，相较于早期工作可实现略微更优的权衡。我们将启发式分析推广至部分早期工作，并与其中非启发式分析进行了比较。