Quantum bounds for 2D-grid and Dyck language

Abstract

We study the quantum query complexity of two problems. First, we consider the problem of determining whether a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most k. We call this the ${{\,\mathrm{\textsc {Dyck}}\,}}_{k,n}$ problem. We prove a lower bound of $\Omega (c^k \sqrt{n})$, showing that the complexity of this problem increases exponentially in k. Here n is the length of the word. When k is a constant, 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. (Electron Colloquium Comput Complex (ECCC) 26:61, 2018). Their proof does not give rise to a general algorithm. When k is not a constant, ${{\,\mathrm{\textsc {Dyck}}\,}}_{k,n}$ is not context-free. We give an algorithm with $O\left( \sqrt{n}(\log {n})^{0.5k}\right)$ quantum queries for ${{\,\mathrm{\textsc {Dyck}}\,}}_{k,n}$ for all k. This is better than the trivial upper bound n for $k=o\left( \frac{\log (n)}{\log \log n}\right)$. Second, we consider connectivity problems on grid graphs 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. We present two algorithms for particular cases of the problem. 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