
Quantum Lower and Upper Bounds for 2DGrid and Dyck Language
We study the quantum query complexity of two problems. First, we consi...
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Quantum Query Complexity of Dyck Languages with Bounded Height
We consider the problem of determining if a sequence of parentheses is w...
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Lower Bounds for Function Inversion with Quantum Advice
Function inversion is that given a random function f: [M] → [N], we want...
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Automation Strategies for Unconstrained Crossword Puzzle Generation
An unconstrained crossword puzzle is a generalization of the constrained...
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Sparsification of Balanced Directed Graphs
Sparsification, where the cut values of an input graph are approximately...
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The Connected Domination Number of Grids
Closed form expressions for the domination number of an n × m grid have ...
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Lower bound the Tcount via unitary stabilizer nullity
We introduce magic measures for multiqubit quantum gates and establish ...
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Quantum Lower Bounds for 2DGrid and Dyck Language
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, Õ(√(n)) queries suffice, with a Õ term depending on k. We prove a lower bound of Ω(c^k √(n)), showing that the complexity of this problem increases exponentially in k. This is interesting as a representative example of starfree languages for which a surprising Õ(√(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 Ω(n^1.5ϵ) for the directed 2D grid and Ω(n^2ϵ) for the undirected 2D grid. The directed problem is interesting as a blackbox model for a class of classical dynamic programming strategies including the one that is usually used for the wellknown edit distance problem. We also show a generalization of this result to more than 2 dimensions.
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