Quantum divide and conquer
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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^aux(n), to give a recurrence relation C(n) ≤ a C(n/b) + C^aux(n) for the classical complexity C(n). We describe a quantum divide-and-conquer framework that, in certain cases, yields an analogous recurrence relation C_Q(n) ≤√(a) C_Q(n/b) + O(C^aux_Q(n)) 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 regular languages; (ii) decision versions of String Rotation and String Suffix; and natural parameterized versions of (iii) Longest Increasing Subsequence and (iv) Longest Common Subsequence.
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