For many problems, quantum algorithms promise speedups over their classical counterparts. However, these results predominantly rely on asymptotic worst-case analysis, which overlooks significant overheads due to error correction and the fact that real-world instances often contain exploitable structure. In this work, we employ the hybrid benchmarking method to evaluate the potential of quantum Backtracking and Grover's algorithm against the 2023 SAT competition main track winner in solving random $k$-SAT instances with tunable structure, designed to represent industry-like scenarios, using both $T$-depth and $T$-count as cost metrics to estimate quantum run times. Our findings reproduce the results of Campbell, Khurana, and Montanaro (Quantum '19) in the unstructured case using hybrid benchmarking. However, we offer a more sobering perspective in practically relevant regimes: almost all quantum speedups vanish, even asymptotically, when minimal structure is introduced or when $T$-count is considered instead of $T$-depth. Moreover, when the requirement is for the algorithm to find a solution within a single day, we find that only Grover's algorithm has the potential to outperform classical algorithms, but only in a very limited regime and only when using $T$-depth. We also discuss how more sophisticated heuristics could restore the asymptotic scaling advantage for quantum backtracking, but our findings suggest that the potential for practical quantum speedups in more structured $k$-SAT solving will remain limited.

翻译：对于许多问题，量子算法有望相较于经典算法实现加速。然而，这些结果主要依赖于渐近最坏情况分析，忽略了纠错带来的显著开销，以及现实世界实例通常包含可利用的结构这一事实。在本工作中，我们采用混合基准测试方法，评估量子回溯算法和Grover算法相对于2023年SAT竞赛主赛道优胜者在求解具有可调结构的随机$k$-SAT实例（设计用于模拟工业场景）时的潜力，同时使用$T$深度和$T$计数作为成本指标来估算量子运行时间。我们的研究结果在非结构化情况下通过混合基准测试复现了Campbell、Khurana和Montanaro（Quantum '19）的结论。然而，在实际相关机制中，我们提出了一个更为审慎的观点：当引入最小结构或考虑$T$计数而非$T$深度时，几乎所有的量子加速优势都会消失，即使在渐近意义上也是如此。此外，当要求算法在一天内找到解时，我们发现只有Grover算法有可能超越经典算法，但仅限于非常有限的范围，并且仅在使用$T$深度时成立。我们还讨论了更复杂的启发式方法如何可能恢复量子回溯算法的渐近标度优势，但我们的研究结果表明，在更具结构性的$k$-SAT求解中，实现实用的量子加速潜力仍然有限。