On the quantum time complexity of divide and conquer
Jonathan Allcock, Jinge Bao, Aleksandrs Belovs, Troy Lee, Miklos Santha
TL;DR
The paper develops a unified framework to analyze the time complexity of quantum divide-and-conquer algorithms for classical problems, introducing generic theorems that convert many DC procedures into quantum-time bounds under specific create/complete-step conditions and memory models. It applies these results to a broad set of problems, including Longest Distinct Substring, Klee's Coverage, and various sequence/substring optimizations, achieving near-optimal time up to polylog factors relative to quantum lower bounds. It also extends the DC approach to disjunctive/minimizing problems and introduces BLDS to handle nontrivial create steps, yielding sublinear time algorithms for LDS and related issues. Finally, it explores quantum complexity for APSP-related rectangle problems (MSM and M4C), delivering both upper bounds and lower-bound insights, and illustrating the nuanced landscape of quantum speedups beyond naive squaring of memory or input size.
Abstract
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.