3 research outputs found
Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
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 ?(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. [Scott Aaronson et al., 2018]. 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(log n)^{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)}/{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 ?(n^{1.5-?}) for the directed 2D grid and ?(n^{2-?}) 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
Quantum Speedups for Treewidth
In this paper, we study quantum algorithms for computing the exact value of
the treewidth of a graph. Our algorithms are based on the classical algorithm
by Fomin and Villanger (Combinatorica 32, 2012) that uses time and
polynomial space. We show three quantum algorithms with the following
complexity, using QRAM in both exponential space algorithms:
time and polynomial space; time and
space; time and space. In contrast, the
fastest known classical algorithm for treewidth uses time and
space. The first two speed-ups are obtained in a fairly straightforward way.
The first version uses additionally only Grover's search and provides a
quadratic speedup. The second speedup is more time-efficient and uses both
Grover's search and the quantum exponential dynamic programming by Ambainis et
al. (SODA '19). The third version uses the specific properties of the classical
algorithm and treewidth, with a modified version of the quantum dynamic
programming on the hypercube. Lastly, as a small side result, we also give a
new classical time-space tradeoff for computing treewidth in time
and space