arXiv subjects

Mark Heiligman

Publications and source records attributed to Mark Heiligman.

7 recordsLinked to original sources

arXiv2015

Mult-Projective Models for Jacobian Varieties of Genus Two Curves

A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to write down affine glue equations for the overlaps. Taking multi-projective completions for all of these then yields a complete multi-projective two-dimensional variety whose points are in one-to-one correspondence with degree zero divisor classes on the curve (i.e. the Picard group). This multi-projective variety then becomes a complete projective two-dimensional variety under the Segre imbedding.

math.AG↗

arXiv2011

Quantum Verification of Minimum Spanning Tree

Previous studies has shown that for a weighted undirected graph having $n$ vertices and $m$ edges, a minimal weight spanning tree can be found with $O^*(\sqrt{mn})$ calls to the weight oracle. The present note shows that a given spanning tree can be verified to be a minimal weight spanning tree with only $O(n\bigr)$ calls to the weight oracle and $O(n+\sqrt{m}\log n)$ total work.

quant-ph↗

arXiv2011

Sequentiality Restrictions in Special Relativity

Observers in different inertial frames can see a set of spacelike separated events as occurring in different orders. Various restrictions are studied on the possible orderings of events that can be observed. In 1+1-dimensional spacetime {(1 2 3), (2 3 1), (3 1 2)} is a disallowed set of permutations. In 3+1-dimensional spacetime, any four different permutations on the ordering of n events can be seen by four different observers, and there is a set of four events such that any of the 4!=24 possible orderings can be observed in some inertial reference frame. A more complicated problem is that of five observers and five events, where of the 7,940,751 choices of five distinct elements from S_5 (containing the identity), all but at most one set of permutations can be realized, and it is shown that this remaining case is impossible. For six events and five observers, it is shown that there are at least 7970 cases that are unrealizable, of which at least 294 do not come from the forbidden configuration of five events.

math.CO↗

arXiv2004

Quantum query complexity of some graph problems

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Theta(n^{3/2}) in the matrix model and in Theta(sqrt{nm}) in the array model, while the complexity of Connectivity is also in Theta(n^{3/2}) in the matrix model, but in Theta(n) in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.

quant-ph↗

arXiv2003

Quantum Algorithms for Lowest Weight Paths and Spanning Trees in Complete Graphs

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 leads to more efficient algorithms. In the case of highly asymmetric complete bipartite graphs, simply replacing classical search with quantum search leads to a faster quantum algorithm. A fast quantum algorithm for computing the diameter of a complete graph is also given.

quant-ph↗

arXiv2000

Quantum Algorithms for Element Distinctness

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Hoyer, and Tapp, and imply an O(N^{3/4} log N) quantum upper bound for the element distinctness problem in the comparison complexity model (contrasting with Theta(N log N) classical complexity). We also prove a lower bound of Omega(N^{1/2}) comparisons for this problem and derive bounds for a number of related problems.

quant-ph↗

arXiv2000

Finding Matches between Two Databases on a Quantum Computer

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 $O(N^{3/4}\log N)$ (measured in quantum memory accesses and accesses to each list). The amount of quantum memory required is $O(N^{1/2})$. The quantum algorithm that accomplishes this consists of an inner Grover search combined with a partial sort all sitting inside of an outer Grover search.

quant-ph↗