Quantum lower and upper bounds for 2D-grid and dyck language

Abstract

© Nathalie Bertrand; licensed under Creative Commons License CC-BY 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). We study the quantum query complexity of 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. We call this the Dyckk,n problem. We prove a lower bound of Ω(ck√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 Õ(√n) query quantum algorithm was recently constructed by Aaronson et al. [1]. Their proof does not give rise to a general algorithm. When k is not a constant, Dyckk,n is not context-free. We give an algorithm with O(√n(log n)0.5k) quantum queries for Dyckk,n for all k. This is better than the trival upper bound n for k = o( loglog(n)/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(n1.5−ε) for the directed 2D grid and Ω(n2−ε) 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.