[论文解读] On the Quantum Time Complexity of Divide and Conquer

Jonathan Allcock, Jinge Bao+3 more|arXiv (Cornell University)|Nov 28, 2023

Quantum Computing Algorithms and Architecture被引用 1

一句话总结

本文通过在合并与完成步骤中利用量子搜索和最小/最大值查找，为经典分治算法建立了通用的量子加速。对于最长不同子串、克利覆盖、股票交易优化和k-递增子序列等问题，该方法实现了近乎最优的量子时间复杂度——与已知的量子查询下界相比，仅相差多对数因子。

ABSTRACT

In this work, we initiate a systematic study of the time complexity of quantum divide and conquer (QD&C) algorithms for classical problems, and propose a general framework for their analysis. 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 various problems involving strings, integers, and geometric objects. These include Longest Distinct Substring, Klee's Coverage, several optimization problems on stock transactions, and k-Increasing Subsequence. For most of these problems our quantum time upper bounds match the quantum query lower bounds, up to polylogarithmic factors. We give a structured framework for describing and classifying a wide variety of QD&C algorithms so that quantum speedups can be more easily identified and applied, and prove general statements on QD&C time complexity covering a range of cases, accounting for the time required for all operations. In particular, we explicitly account for memory access operations in the commonly used QRAM (read-only) and QRAG (read-write) models, which are assumed to take unit time in the query model, and which require careful analysis when involved in recursion. Our generic QD&C theorems have several nice features. 1) To apply them, it suffices to come up with a classical divide and conquer algorithm satisfying the conditions of the theorem. The quantization of the algorithm is then completely handled by the theorem. This can make it easier to find applications which admit a quantum speedup, and contrast with dynamic programming algorithms which can be difficult to quantize due to their highly sequential nature. 2) As these theorems give bounds on time complexity, they can be applied to a greater range of problems than those based on query complexity, e.g., where the best-known quantum algorithms require super-linear time. 3) It can handle minimization problems as well as boolean functions, which allows us to improve on the query complexity result of Childs et al. [Childs et al., 2025] for k-Increasing Subsequence by a logarithmic factor.

研究动机与目标

• 系统分析经典分治算法的量子时间复杂度。
• 识别此类算法在何种通用条件下可实现量子加速。
• 构建一个理论框架，将经典分治算法转化为具有可证明时间复杂度边界的量子版本。
• 将该框架应用于字符串、整数和计算几何中的具体问题。
• 建立紧致的量子时间上界，使其与已知的量子查询下界相比仅相差多对数因子。

提出的方法

• 提出一个四步框架——创建、征服、完成、合并——以建模分治算法。
• 在合并与完成步骤中应用量子搜索（Grover）和量子最小/最大值查找。
• 使用QRAM和QRAG量子内存模型，以建模对经典数据的相干访问及叠加内存。
• 利用量子查询复杂度和动态规划技术推导时间复杂度上界。
• 通过布尔多数问题的约化，证明不可约的量子查询下界。
• 将已知的1D问题（如单只股票单次交易）的量子算法作为高维推广的子程序使用。

实验结果

研究问题

• RQ1 在何种条件下，经典分治算法可通过量子计算实现加速？
• RQ2 是否可以系统性地将子问题的量子加速通过通用变换传递到完整问题？
• RQ3 在量子内存模型下，最长不同子串和克利覆盖问题的量子时间复杂度是多少？
• RQ4 能否建立与所提量子算法上界匹配的量子查询下界？
• RQ5 APSP类中是否存在量子时间复杂度不属于n^{3/2}或n^{5/2}阶的问题？该问题能否通过量子分治解决？

主要发现

• 在QRAM模型下，最长不同子串问题的量子时间复杂度为Õ(n^{2/3})，与量子查询下界相比仅相差多对数因子。
• 对于d维空间中d ≥ 8的克利覆盖问题，量子时间复杂度为O(n^{d/4 + ε})，优于经典O(n^{d/2})的复杂度。
• 单只股票单次交易问题的量子时间复杂度为Õ(√n log^{5/2} n)，与已知的量子查询下界一致。
• k-递增子序列问题的量子时间复杂度为Õ(n)，与量子查询下界相比仅相差多对数因子。
• 最大子矩阵问题（MSM）在QRAG模型下的量子时间复杂度为O(n² log² n)，并具有紧致的Ω(n²)量子查询下界。
• 最大四组合问题（M4C）在QRAM模型下的量子时间复杂度为O(n^{3/2} log^{5/2} n)，暗示对相关问题（如最大三角形）可能存在亚n^{3/2}阶的量子算法。