We initiate a systematic study of the time complexity of quantum divide and conquer algorithms for classical problems. We establish generic conditions under which search and minimization problems with classical divide and conquer algorithms are amenable to quantum speedup and apply these theorems to an array of problems involving strings, integers, and geometric objects. They include LONGEST DISTINCT SUBSTRING, KLEE'S COVERAGE, several optimization problems on stock transactions, and k-INCREASING SUBSEQUENCE. For most of these results, our quantum time upper bound matches the quantum query lower bound for the problem, up to polylogarithmic factors.

翻译：我们首次系统研究了经典问题中量子分治算法的时间复杂度。针对具有经典分治算法的搜索和最小化问题，我们建立了可量子加速的通用条件，并将这些定理应用于涉及字符串、整数和几何对象的系列问题，包括最长不重复子串、KLEE覆盖、多个股票交易优化问题以及k-递增子序列。对于大部分结果，我们的量子时间上界与问题对应的量子查询下界匹配，仅相差多对数因子。