Mark Heiligman

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We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N3=4 logN) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with (N logN) classical(More)
Quantum algorithms for several problems in graph theory are considered. Classical algorithms for finding the lowest weight path between two points in a graph and for finding a minimal weight spanning tree involve searching over some space. Modification of classical algorithms due to Dijkstra and Prim allows quantum search to replace classical search and(More)
Given two unsorted lists each of length N that have a single common entry, a quantum computer can find that matching element with a work factor of O(N logN) (measured in quantum memory accesses and accesses to each list). The amount of quantum memory required is O(N). The quantum algorithm that accomplishes this consists of an inner Grover search combined(More)
<i>Weierstrass points</i> are an important geometric feature and birational invariant of an algebraic curve. A place <i>P</i> on a curve of genus <i>g</i> is a Weierstrass point if the corresponding point on the canonical embedding of the curve is an inflection point where the tangent plane is tangent to order at least <i>g.</i> This condition is equivalent(More)
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