The traveling salesman problem (TSP) is a significant classical NP-hard combinatorial optimization problem. In this work, we demonstrate that combining classical dynamic programming with quantum search can yield an achievable quantum advantage for TSP on the basis of excellent work by the authors of~\cite{ambainis2019quantum}. We design the quantum divide and conquer strategy to provide a parameterized spectrum for this combination. The hybrid algorithm proposed in~\cite{ambainis2019quantum} corresponds to a specific case in this spectrum, while the two extremes of the spectrum represent the purely classical Held-Karp and the purely quantum search algorithm, respectively. Within our parameterized spectrum, we prove that the optimal query complexity is $O^*(1.865666\ldots^n)$, achieved with the 4-subset scheme, while the counting in~\cite{ambainis2019quantum} overlooked half of the recursive branches. The correct query complexity of their algorithm is $O^*(2.225880\ldots^n)$ at their chosen parameter ($α\approx0.055362$), and cannot fall below $O^*(2^n)$ for any $α$ - meaning their $8$-subset scheme, correctly analyzed, never surpasses the classical Held-Karp bound. Furthermore, in previous studies on quantum advantages for NP-hard combinatorial optimization problems, researchers focused only on improvements in query complexity. Our work, however, points out that the quantum advantage stems not only from the quadratic speedup of quantum search but also from the structured quantum state preparation. We argue that structured state preparation is indispensable for realizing the oracle operator while maintaining the total time complexity of $O^*(1.865666\ldots^n)$. Therefore, we design an elegant method for preparing the set partition state, which makes our TSP solver practically executable.

翻译：旅行商问题（TSP）是一类重要的经典NP-困难组合优化问题。本文基于文献~\cite{ambainis2019quantum}的卓越工作，证明将经典动态规划与量子搜索相结合可在TSP问题上实现可获取的量子优势。我们设计了量子分治策略，为该组合方法提供了参数化谱系。文献~\cite{ambainis2019quantum}提出的混合算法对应谱系中的特定情形，而谱系的两个极端分别代表纯经典Held-Karp算法与纯量子搜索算法。在该参数化谱系内，我们证明最优查询复杂度为$O^*(1.865666\ldots^n)$，可通过4-子集方案实现，而文献~\cite{ambainis2019quantum}的计数遗漏了半数递归分支。在其选定参数（$α\approx0.055362$）下，该算法的正确查询复杂度为$O^*(2.225880\ldots^n)$，且对任意$α$均无法低于$O^*(2^n)$——这意味着其8-子集方案经正确分析后始终无法超越经典Held-Karp界。此外，在关于NP-困难组合优化问题量子优势的既往研究中，研究者仅关注查询复杂度的改进。而本文指出，量子优势不仅来源于量子搜索的二次加速，更源自结构化量子态的制备。我们论证在保持总时间复杂度为$O^*(1.865666\ldots^n)$的前提下，结构化态制备对实现Oracle算子不可或缺。为此，我们设计了一种优雅的集合分割态制备方法，使得我们的TSP求解器具备实际可执行性。