The short-path quantum algorithm introduced by Hastings (Quantum 2018, 2019) is a variant of adiabatic quantum algorithms that enables an easier worst-case analysis by avoiding the need to control the spectral gap along a long adiabatic path. Dalzell, Pancotti, Campbell, and Brandão (STOC 2023) recently revisited this framework and obtained a clear analysis of the complexity of the short-path algorithm for several classes of constraint satisfaction problems (MAX-$k$-CSPs), leading to quantum algorithms with complexity $2^{(1-c)n/2}$ for some constant $c>0$. This suggested a super-quadratic quantum advantage over classical algorithms. In this work, we identify an explicit classical mechanism underlying a substantial part of this line of work, and show that it leads to clean dequantizations. As a consequence, we obtain classical algorithms that run in time $2^{(1-c')n}$, for some constant $c'>c$, for the same classes of constraint satisfaction problems. This shows that current short-path quantum algorithms for these problems do not achieve a super-quadratic advantage. On the positive side, our results provide a new ``quantum-inspired'' approach to designing classical algorithms for important classes of constraint satisfaction problems.

翻译：Hastings（量子 2018, 2019）提出的短程路径量子算法是绝热量子算法的一种变体，它通过避免控制长绝热路径上的能隙，简化了最坏情况分析。Dalzell、Pancotti、Campbell 和 Brandão（STOC 2023）最近重新审视了这一框架，并针对几类约束满足问题（MAX-$k$-CSPs）获得了短程路径算法复杂性的清晰分析，从而得到了复杂度为 $2^{(1-c)n/2}$（其中 $c>0$ 为常数）的量子算法。这表明量子算法相对于经典算法具有超二次优势。本文中，我们识别出这类工作很大一部分背后的显式经典机制，并展示其导致清晰的反量子化。因此，对于同一类约束满足问题，我们获得了运行时间为 $2^{(1-c')n}$（其中常数 $c'>c$）的经典算法。这表明目前针对这些问题的短程路径量子算法并未实现超二次优势。从积极方面看，我们的结果为重要约束满足问题类的经典算法设计提供了一种新的“量子启发式”方法。