
Range closestpair search in higher dimensions
Range closestpair (RCP) search is a rangesearch variant of the classic...
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The Quantum Strong ExponentialTime Hypothesis
The strong exponentialtime hypothesis (SETH) is a commonly used conject...
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Multidimensional Included and Excluded Sums
This paper presents algorithms for the includedsums and excludedsums p...
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Faster quantum and classical SDP approximations for quadratic binary optimization
We give a quantum speedup for solving the canonical semidefinite program...
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Depthscaling finegrained quantum supremacy based on SETH and qubitscaling finegrained quantum supremacy based on Orthogonal Vectors and 3SUM
We first show that under SETH and its variant, strong and weak classical...
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An Equivalence Class for Orthogonal Vectors
The Orthogonal Vectors problem (OV) asks: given n vectors in {0,1}^O( n)...
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Quantum algorithms for computational geometry problems
We study quantum algorithms for problems in computational geometry, such...
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On the Quantum Complexity of Closest Pair and Related Problems
The closest pair problem is a fundamental problem of computational geometry: given a set of n points in a ddimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in O(nlog n) time in constant dimensions (i.e., when d=O(1)). This paper asks and answers the question of the problem's quantum time complexity. Specifically, we give an O(n^2/3) algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient historyindependent data structure that supports quantum interference. In polylog(n) dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover's algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum finegrained complexity and introduce the Quantum Strong Exponential Time Hypothesis (QSETH), which is based on the assumption that Grover's algorithm is optimal for CNFSAT when the clause width is large. We show that the naïve Grover approach to closest pair in higher dimensions is optimal up to an n^o(1) factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.
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