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 $\tilde{\Theta}(\sqrt{n})$, where the hidden logarithm factors in $\tilde{\Theta}$ 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 $0<\epsilon\leq 0.37$, there exists $c>0$ such that $Q(\text{Dyck}_{c\log(n)}(n))=\Omega(n^{1-\epsilon})$. When $h=\omega(\log(n))$, the quantum query complexity is close to $n$, i.e. $Q(\text{Dyck}_h(n))=\omega(n^{1-\epsilon})$ for all $\epsilon>0$. Furthermore when $h=\Omega(n^\epsilon)$ for some $\epsilon>0$, $Q(\text{Dyck}_{h}(n))=\Theta(n)$.