Run-times of quantum algorithms are often studied via an asymptotic, worst-case analysis. Whilst useful, such a comparison can often fall short: it is not uncommon for algorithms with a large worst-case run-time to end up performing well on instances of practical interest. To remedy this it is necessary to resort to run-time analyses of a more empirical nature, which for sufficiently small input sizes can be performed on a quantum device or a simulation thereof. For larger input sizes, alternative approaches are required. In this paper we consider an approach that combines classical emulation with detailed complexity bounds that include all constants. We simulate quantum algorithms by running classical versions of the sub-routines, whilst simultaneously collecting information about what the run-time of the quantum routine would have been if it were run instead. To do this accurately and efficiently for very large input sizes, we describe an estimation procedure and prove that it obtains upper bounds on the true expected complexity of the quantum algorithms. We apply our method to some simple quantum speedups of classical heuristic algorithms for solving the well-studied MAX-$k$-SAT optimization problem. This requires rigorous bounds (including all constants) on the expected- and worst-case complexities of two important quantum sub-routines: Grover search with an unknown number of marked items, and quantum maximum-finding. These improve upon existing results and might be of broader interest. Amongst other results, we found that the classical heuristic algorithms we studied did not offer significant quantum speedups despite the existence of a theoretical per-step speedup. This suggests that an empirical analysis such as the one we implement in this paper already yields insights beyond those that can be seen by an asymptotic analysis alone.

翻译：量子算法的运行时间通常通过渐近最坏情况分析来研究。虽然这种分析有用，但往往存在不足：一些最坏情况运行时间较大的算法在实际感兴趣的问题上却表现良好并不罕见。为解决这一问题，有必要采用更偏经验性质的运行时分析，对于足够小的输入规模，可以在量子设备或其模拟器上执行。对于更大的输入规模，则需要替代方法。本文考虑一种将经典模拟与包含所有常数的详细复杂度界限相结合的方法。我们通过运行子程序的经典版本来模拟量子算法，同时收集关于如果运行量子程序本应产生的运行时信息。为了对非常大的输入规模准确且高效地执行此操作，我们描述了一种估计过程，并证明它能够获得量子算法真实期望复杂度的上界。我们将该方法应用于解决经典MAX-$k$-SAT优化问题的一些简单量子加速经典启发式算法。这需要两个重要量子子程序（具有未知数量标记项的Grover搜索和量子最大值查找）的期望和最坏情况复杂度的严格界限（包括所有常数）。这些结果改进了现有研究，并可能具有更广泛的意义。在其他结果中，我们发现所研究的经典启发式算法尽管存在理论上的每步加速，但并未提供显著的量子加速。这表明，本文实施的此类经验分析能提供超越单独渐近分析所能看到的见解。