Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
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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 Dyck_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 Õ(√(n)) query quantum algorithm was recently constructed by Aaronson et al. Their proof does not give rise to a general algorithm. When k is not a constant, Dyck_k,n is not context-free. We give an algorithm with O(√(n)(logn)^0.5k) quantum queries for Dyck_k,n for all k. This is better than the trival upper bound n for k=o(log(n)/loglog 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. 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.
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