Quantum Query Complexity of Dyck Languages with Bounded Height

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 $\tildeΘ(\sqrt{n})$, where the hidden logarithm factors in $\tildeΘ$ 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 $O\left(\sqrt{n}\log(n)^{0.5h}\right)$ quantum queries for $Dyck_h$ for all h. This is better than the trival upper bound $n$ when $h=o(\frac{\log(n)}{\log\log n})$. We also obtain lower bounds: we show that for every $0<ε\leq 0.37$, there exists $c>0$ such that $Q(\text{Dyck}_{c\log(n)}(n))=Ω(n^{1-ε})$. When $h=ω(\log(n))$, the quantum query complexity is close to $n$, i.e. $Q(\text{Dyck}_h(n))=ω(n^{1-ε})$ for all $ε>0$. Furthermore when $h=Ω(n^ε)$ for some $ε>0$, $Q(\text{Dyck}_{h}(n))=Θ(n)$.

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