Quantum Lower Bounds for 2D-Grid and Dyck Language

TL;DR

This paper establishes quantum lower bounds for the complexity of checking balanced parentheses with bounded depth and for connectivity problems in 2D grids with missing edges, revealing exponential and polynomial lower bounds respectively.

Contribution

It proves exponential quantum lower bounds for Dyck language recognition with bounded depth and polynomial bounds for grid connectivity problems, advancing understanding of quantum query complexity.

Findings

Quantum lower bound of  c^k \u221a{n} for Dyck language with depth k

Lower bound of  n^{1.5-\u03b5} for directed 2D grid connectivity

Lower bound of  n^{2-} for undirected 2D grid connectivity

Abstract

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.