Quantum bounds for 2D-grid and Dyck language

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 D Y C K k , n problem. We prove a lower bound of Ω ( c k 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 O ~ ( 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, D Y C K k , n is not context-free. We give an algorithm with O n ( log n ) 0.5 k quantum queries for D Y C K k , n for all k . This is better than the trivial upper bound n for k = o log ( n ) log log n . 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 Ω ( n 1.5 - ϵ ) for the directed 2D grid and Ω ( n 2 - ϵ ) 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