Quantum Divide and Conquer

The divide-and-conquer framework, used extensively in classical algorithm design, recursively breaks a problem of size n into smaller subproblems (say, a copies of size \(n/b\) each), along with some auxiliary work of cost \(C^{\mathrm{aux}}(n)\) , to give a recurrence relation \(\begin{equation*} C(n) \le a \, C(n/b) + C^{\mathrm{aux}}(n) \end{equation*}\) for the classical complexity \(C(n)\) . We describe a quantum divide-and-conquer framework that, in certain cases, yields an analogous recurrence relation \(\begin{equation*} C_Q(n) \le \sqrt {a} \, C_Q(n/b) + O(C^{\mathrm{aux}}_Q(n)) \end{equation*}\) that characterizes the quantum query complexity. We apply this framework to obtain near-optimal quantum query complexities for various string problems, such as (i) recognizing the regular language \(\Sigma ^* 2 0^* 2 \Sigma ^*\) over the alphabet \(\Sigma = \lbrace 0,1,2\rbrace\) ; (ii) decision versions of String Rotation and String Suffix; and natural parameterized versions of (iii) Longest Increasing Subsequence and (iv) Longest Common Subsequence.