Quantum Query Complexity of Dyck Languages with Bounded Height
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We consider the problem of determining if a sequence of parentheses is well parenthesized, with a depth of at most h. We denote this language as Dyck_h. We study the quantum query complexity of this problem for different h as function of the length n of the word. It has been known from a recent paper by Aaronson et al. that, for any constant h, since Dyck_h is star-free, it has quantum query complexity Θ̃(√(n)), where the hidden logarithm factors in Θ̃ depend on h. Their proof does not give rise to an algorithm. When h is not a constant, Dyck_h is not even context-free. We give an algorithm with O(√(n)log(n)^h-1) quantum queries for Dyck_h for all h. This is better than the trival upper bound n when h=o(log(n)/loglog n). We also obtain lower bounds: we show that for every 0<ϵ≤ 0.37, there exists c>0 such that Q(Dyck_clog(n)(n))=Ω(n^1-ϵ). When h=ω(log(n)), the quantum query complexity is close to n, i.e. Q(Dyck_h(n))=ω(n^1-ϵ) for all ϵ>0. Furthermore when h=Ω(n^ϵ) for some ϵ>0, Q(Dyck_h(n))=Θ(n).
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