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Characterization of non-universal two-qubit HamiltoniansMancinska, Laura January 2009 (has links)
It is known that almost all 2-qubit gates are universal for quantum computing (Lloyd 1995; Deutsch, Barenco, Eckert 1995). However, an explicit characterization of non-universal 2-qubit gates is not known. We consider a closely related problem of characterizing the set of non-universal 2-qubit Hamiltonians. We call a 2-qubit Hamiltonian n-universal if, when applied on different pairs of qubits, it can be used to approximate any unitary operation on n qubits. It follows directly from the results of Lloyd and Deutsch, Barenco, Eckert, that almost any 2-qubit Hamiltonian is 2-universal. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. There are three cases when a 2-qubit Hamiltonian H is not universal:
(1) H shares an eigenvector with the gate that swaps two qubits;
(2) H acts on the two qubits independently (in any of a certain family of bases);
(3) H has zero trace.
The last condition rules out the Hamiltonians that generate SU(4)---it can be omitted if the global phase is not important.
A Hamiltonian that is not 2-universal can still be 3-universal. We give a (possibly incomplete) list of 2-qubit Hamiltonians that are not 3-universal. If this list happens to be complete, it actually gives a classification of n-universal 2-qubit Hamiltonians for all n >= 3.
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Exact, Approximate, and Online Algorithms for Optimization Problems Arising in DVD AssignmentPearson, James Ross January 2009 (has links)
Zip.ca is an online DVD rental company that faces two major operational problems:
calculation of the assignment of DVDs to customers every thirty minutes
throughout the day and purchasing of new inventory in regular intervals.
In this thesis, we model these two problems and develop algorithms to solve
them. In doing so, we encounter many theoretical problems that are both applicable
to Zip’s operations and intrinsically interesting problems independent of the
application.
First, we note that the assignment problem facing Zip is inherently in an online
setting. With returns of DVDs being processed throughout the day, the dataset
is constantly changing. Although the ideal solution would be to wait until the
end of the day to make decisions, physical work load capacities prevent this. For
this reason we discuss two online problems, online 0-1 budgeted matching and
the budgeted Adwords auction. We present a 1/(2 w_max/w_min)-competitive algorithm for the online 0-1 budgeted matching problem, and prove that this is the best possible
competitive ratio possible for a wide class of algorithms. We also give a (1− (S+1)/(S+e) )-competitive algorithm for the budgeted Adwords auction as the size of the bids
and cost get small compared to the budgets, where S is the ratio of the highest and
lowest ratios of bids to costs.
We suggest a linear programming approach to solve Zip’s assignment problem.
We develop an integer program that models the B-matching instance with additional
constraints of concern to Zip, and prove that this integer program belongs to
a larger class of integer programs that has totally unimodular constraint matrices.
Thus, the assignment problem can be solved to optimality every thirty minutes.
We additionally create a test environment to check daily performance, and provide
real-time implementation results, showing a marked improvement over Zip’s old
algorithm.
We show that Zip’s purchasing problem can be modeled by the matching augmentation
problem defined as follows. Given a graph with vertex capacities and
costs, edge weights, and budget C, find a purchasing of additional node capacity of
cost at most C that admits a B-matching of maximum weight. We give a PTAS for
this problem, and then present a special case that is polynomial time solvable that
still models Zip’s purchasing problem, under the assumption of uniform costs.
We then extend the augmentation idea to matroids and present matroid augmentation,
matroid knapsack, and matroid intersection knapsack, three NP-hard
problems. We give an FPTAS for matroid knapsack by dynamic programming,
PTASes for the other two, and demonstrate applications of these problems.
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Combinatorics and the KP HierarchyCarrell, Sean January 2009 (has links)
The study of the infinite (countable) family of partial differential equations
known as the Kadomtzev - Petviashvili (KP) hierarchy has received much interest in
the mathematical and theoretical physics community for over forty years. Recently
there has been a renewed interest in its application to enumerative combinatorics
inspired by Witten's conjecture (now Kontsevich's theorem).
In this thesis we provide a detailed development of the KP hierarchy and some of
its applications with an emphasis on the combinatorics involved. Up until now, most
of the material pertaining to the KP hierarchy has been fragmented throughout the
physics literature and any complete accounts have been for purposes much diff erent
than ours.
We begin by describing a family of related Lie algebras along with a module
on which they act. We then construct a realization of this module in terms of
polynomials and determine the corresponding Lie algebra actions. By doing this
we are able to describe one of the Lie group orbits as a family of polynomials and the
equations that de fine them as a family of partial diff erential equations. This then
becomes the KP hierarchy and its solutions. We then interpret the KP hierarchy
as a pair of operators on the ring of symmetric functions and describe their action
combinatorially. We then conclude the thesis with some combinatorial applications.
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Generation and properties of random graphs and analysis of randomized algorithmsGao, Pu January 2010 (has links)
We study a new method of generating random $d$-regular graphs by
repeatedly applying an operation called pegging. The pegging
algorithm, which applies the pegging operation in each step, is a
method of generating large random regular graphs beginning with
small ones. We prove that the limiting joint distribution of the
numbers of short cycles in the resulting graph is independent
Poisson. We use the coupling method to bound the total variation
distance between the joint distribution of short cycle counts and
its limit and thereby show that $O(\epsilon^{-1})$ is an upper bound
of the $\eps$-mixing time. The coupling involves two different,
though quite similar, Markov chains that are not time-homogeneous.
We also show that the $\epsilon$-mixing time is not
$o(\epsilon^{-1})$. This demonstrates that the upper bound
is essentially tight. We study also the
connectivity of random $d$-regular graphs generated by the pegging
algorithm. We show that these graphs are asymptotically almost
surely $d$-connected for any even constant $d\ge 4$.
The problem of orientation of random hypergraphs is motivated by the
classical load balancing problem. Let $h>w>0$ be two fixed integers.
Let $\orH$ be a hypergraph whose hyperedges are uniformly of size
$h$.
To {\em $w$-orient} a hyperedge, we assign exactly $w$ of its
vertices positive signs with respect to this hyperedge, and the rest
negative. A $(w,k)$-orientation of $\orH$ consists of a
$w$-orientation of all hyperedges of $\orH$, such that each vertex
receives at most $k$ positive signs from its incident hyperedges.
When $k$ is large enough, we determine the threshold of the
existence of a $(w,k)$-orientation of a random hypergraph. The
$(w,k)$-orientation of hypergraphs is strongly related to a general
version of the off-line load balancing problem.
The other topic we discuss is computing the probability of induced
subgraphs in a random regular graph. Let $0<s<n$ and $H$ be a graph
on $s$ vertices. For any $S\subset [n]$ with $|S|=s$, we compute the
probability that the subgraph of $\mathcal{G}_{n,d}$ induced by $S$
is $H$. The result holds for any $d=o(n^{1/3})$ and is further
extended to $\mathcal{G}_{n,{\bf d}}$, the probability space of
random graphs with given degree sequence $\bf d$. This result
provides a basic tool for studying properties, for instance the
existence or the counts, of certain types of induced subgraphs.
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A New Class of Cycle Inequality for the Time-Dependent Traveling Salesman ProblemWhite, John Lincoln January 2010 (has links)
The Time-Dependent Traveling Salesman Problem is a generalization of the well-known Traveling Salesman Problem, where the cost for travel between two nodes is dependent on the nodes and their position in the tour. Inequalities for the Asymmetric TSP can be easily extended to the TDTSP, but the added time information can be used to strengthen these inequalities. We look at extending the Lifted Cycle Inequalities, a large family of inequalities for the ATSP. We define a new inequality, the Extended Cycle (X-cycle) Inequality, based on cycles in the graph. We extend the results of Balas and Fischetti for Lifted Cycle Inequalities to define Lifted X-cycle Inequalities. We show that the Lifted X-cycle Inequalities include some inequalities which define facets of the submissive of the TDTS Polytope.
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Evaluating Large Degree Isogenies between Elliptic CurvesSoukharev, Vladimir 12 1900 (has links)
An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves efficiently. For ordinary curves of the same endomorphism ring, the previous fastest algorithm known has a worst case running time which is exponential in the length of the input. In this thesis we solve this problem in subexponential time under reasonable heuristics. We give two versions of our algorithm, a slower version assuming GRH and a faster version assuming stronger heuristics. Our approach is based on factoring the ideal corresponding to the kernel of the isogeny, modulo principal ideals, into a product of smaller prime ideals for which the isogenies can be computed directly. Combined with previous work of Bostan et al., our algorithm yields equations for large degree isogenies in quasi-optimal time given only the starting curve and the kernel.
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Infinite graphs, graph-like spaces and B-matroidsChristian, Robin January 2010 (has links)
The central theme of this thesis is to prove results about infinite mathematical objects by studying the behaviour of their finite substructures.
In particular, we study B-matroids, which are an infinite generalization of matroids introduced by Higgs \cite{higgs}, and graph-like spaces, which are topological
spaces resembling graphs, introduced by Thomassen and Vella \cite{thomassenvella}.
Recall that the circuit matroid of a finite graph is a matroid defined on the edges of the graph, with a set of edges being independent if it contains
no circuit. It turns out that graph-like continua and infinite graphs both have circuit B-matroids. The first main result of this thesis is a generalization of
Whitney's Theorem that a graph has an abstract dual if and only if it is planar. We show that an infinite graph has an abstract dual (which is a graph-like
continuum) if and only if it is planar, and also that a graph-like continuum has an abstract dual (which is an infinite graph) if and only if it is planar.
This generalizes theorems of Thomassen (\cite{thomassendual}) and Bruhn and Diestel (\cite{bruhndiestel}). The difficult part of the proof is extending
Tutte's characterization of graphic matroids (\cite{tutte2}) to finitary or co-finitary B-matroids. In order to prove this characterization, we introduce a technique for
obtaining these B-matroids as the limit of a sequence of finite minors.
In \cite{tutte}, Tutte proved important theorems about the peripheral (induced and non-separating) circuits of a $3$-connected graph. He showed that for
any two edges of a $3$-connected graph there is a peripheral circuit containing one but not the other, and that the peripheral circuits of a $3$-connected
graph generate its cycle space. These theorems were generalized to $3$-connected binary matroids by Bixby and Cunningham (\cite{bixbycunningham}).
We generalize both of these theorems to $3$-connected binary co-finitary B-matroids.
Richter, Rooney and Thomassen \cite{richterrooneythomassen} showed that a locally connected, compact metric space has an embedding in the sphere unless it contains a subspace homeomorphic
to $K_5$ or $K_{3,3}$, or one of a small number of other obstructions. We are able to extend this result to an arbitrary surface $\Sigma$; a locally
connected, compact metric space embeds in $\Sigma$ unless it contains a subspace homeomorphic to a finite graph which does not embed in $\Sigma$, or
one of a small number of other obstructions.
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Even Cycle and Even Cut MatroidsPivotto, Irene January 2011 (has links)
In this thesis we consider two classes of binary matroids, even cycle matroids and even cut matroids. They are a generalization of graphic and cographic matroids respectively. We focus on two main problems for these classes of matroids. We first consider the Isomorphism Problem, that is the relation between two representations of the same matroid. A representation of an even cycle matroid is a pair formed by a graph together with a special set of edges of the graph. Such a pair is called a signed graph. A representation for an even cut matroid is a pair formed by a graph together with a special set of vertices of the graph. Such a pair is called a graft. We show that two signed graphs representing the same even cycle matroid relate to two grafts representing the same even cut matroid. We then present two classes of signed graphs and we solve the Isomorphism Problem for these two classes. We conjecture that any two representations of the same even cycle matroid are either in one of these two classes, or are related by a local modification of a known operation, or form a sporadic example. The second problem we consider is finding the excluded minors for these classes of matroids. A difficulty when looking for excluded minors for these classes arises from the fact that in general the matroids may have an arbitrarily large number of representations. We define degenerate even cycle and even cut matroids. We show that a 3-connected even cycle matroid containing a 3-connected non-degenerate minor has, up to a simple equivalence relation, at most twice as many representations as the minor. We strengthen this result for a particular class of non-degenerate even cycle matroids. We also prove analogous results for even cut matroids.
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A survey of Roth's Theorem on progressions of length threeNishizawa, Yui 06 December 2011 (has links)
For any finite set B and a subset A⊆B, we define the density of A in B to be the value α=|A|/|B|. Roth's famous theorem, proven in 1953, states that there is a constant C>0, such that if A⊆{1,...,N} for a positive integer N and A has density α in {1,...,N} with α>C/loglog N, then A contains a non-trivial arithmetic progression of length three (3AP). The proof of this relies on the following dichotomy: either 1) A looks like a random set and the number of 3APs in A is close to the probabilistic expected value, or 2) A is more structured and consequently, there is a progression P of about length α√N on which A∩P has α(1+cα) for some c>0. If 1) occurs, then we are done. If 2) occurs, then we identify P with {1,...,|P|} and repeat the above argument, whereby the density increases at each iteration of the dichotomy. Due to the density increase in case 2), an argument of this type is called a density increment argument. The density increment is obtained by studying the Fourier transforms of the characterstic function of A and extracting a structure out of A. Improving the lower bound for α is still an active area of research and all improvements so far employ a density increment. Two of the most recent results are α>C(loglog N/log N)^{1/2} by Bourgain in 1999 and α>C(loglog N)^5/log N by Sanders in 2010. This thesis is a survey of progresses in Roth's theorem, with a focus on these last two results. Attention was given to unifying the language in which the results are discussed and simplifying the presentation.
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2-crossing critical graphs with a V8 minorAustin, Beth Ann January 2012 (has links)
The crossing number of a graph is the minimum number of pairwise crossings of edges among all planar drawings of the graph. A graph G is k-crossing critical if it has crossing number k and any proper subgraph of G has a crossing number less than k.
The set of 1-crossing critical graphs is is determined by Kuratowski’s Theorem to be {K5, K3,3}. Work has been done to approach the problem of classifying all 2-crossing critical graphs. The graph V2n is a cycle on 2n vertices with n intersecting chords. The only remaining graphs to find in the classification of 2-crossing critical graphs are those that are 3-connected with a V8 minor but no V10 minor.
This paper seeks to fill some of this gap by defining and completely describing a class of graphs called fully covered. In addition, we examine other ways in which graphs may be 2-crossing critical. This discussion classifies all known examples of 3-connected, 2-crossing critical graphs with a V8 minor but no V10 minor.
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