Quantum Lower Bounds for 2D-Grid and Dyck Language

We show quantum lower bounds for two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most $k$. It has been known that, for any $k$, $\tilde{O}(\sqrt{n})$ queries suffice, with a $\tilde{O}$ term depending on $k$. We prove a lower bound of $\Omega(c^k \sqrt{n})$, showing that the complexity of this problem increases exponentially in $k$. This is interesting as a representative example of star-free languages for which a surprising $\tilde{O}(\sqrt{n})$ query quantum algorithm was recently constructed by Aaronson et al. Second, we consider connectivity problems on directed/undirected grid in 2 dimensions, if some of the edges of the grid may be missing. By embedding the "balanced parentheses" problem into the grid, we show a lower bound of $\Omega(n^{1.5-\epsilon})$ for the directed 2D grid and $\Omega(n^{2-\epsilon})$ for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.