Open quantum systems refer to systems that are affected by interaction with the environment. The effects of these unwanted interactions, called \emph{quantum noise}, are studied using dynamical maps. We study the geometry of these maps in this work. We review the canonical representations of dynamical maps such as reduced dynamics, $\mathcal{A}$ and $\mathcal{B}$ forms and operator sum representation. We develop a framework for simplifying the action of dynamical maps in terms of their action on the coherence vector associated with the density matrix. We use the framework to describe the geometry of depolarization, dephasing and dissipation in the domain of complete positivity. We give a geometric picture of how two-, three- and four-level systems are affected by these common forms of quantum noises. We show useful similarities between two- and four-level depolarizing maps and give a generalization for $n$-qubits. We also derive important results that restrict dephasing and dissipation. / text

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One of the most important quantum algorithms is Grover's search algorithm [G96]. Quantum searching can be used to speed up the search for solutions to NP-complete problems e. g. 3SAT. Even so, the best known quantum algorithms for 3SAT are considered inefficient. Soon after Grover's discovery, Farhi and Gutmann [FG96] devised a "continuous-time analogue" of quantum searching. More recently Farhi <i>et. al. </i> [FGGS00] proposed a continuous-time 3SAT algorithm which invokes the adiabatic approximation [M76]. Their algorithm is difficult to analyze, hence we do not know whether it can solve typical 3SAT instances faster than Grover's search algorithm can. I begin with a review of the discrete- and continuous-time models of quantum computation. I then make precise the notion of "efficient quantum algorithms", motivating sufficient conditions for discrete- and continuous-time algorithms to be considered efficient via discussion of standard techniques for discrete-time simulation of continuous-time algorithms. After reviewing three quantum search algorithms [F00,FG96,G96], I develop the adiabatic 3SAT algorithm as a natural extension of Farhi and Gutmann's search algorithm. Along the way, I present the adiabatic search algorithm [vDMV01] and remark on its discrete-time simulation. Finally I devise a generalization of the adiabatic algorithm and prove some lower bounds for various cases of this general framework. UPDATE (February 2003): Please see article http://arxiv. org/abs/quant-ph/0302138 for a resolution to the problem of simulating the continuous-time adiabatic search algorithm with a quantum circuit using only O(sqrt(N)) resources.

A consensus on how to characterise the anterograde and retrograde memory processes that are lost or spared after hippocampal damage has not been reached. In this thesis, I critically re-examine the empirical literature and the assumptions behind current theories. I formulate a coherent view of what makes a task hippocampally dependent at acquisition and how this relates to its long-term fate. Findings from a neural net simulation indicate the plausibility of my proposals. My proposals both extend and constrain current views on the role of the hippocampus in the rapid acquisition of information and in learning complex associations. In general, tasks are most likely to require the hippocampus for acquisition if they involve rapid, associative learning about unfamiliar, complex, low salience stimuli. However, none of these factors alone is sufficient to obligatorily implicate the hippocampus in acquisition. With the exception of associations with supra-modal information that are always dependent on the hippocampus, it is the combination of factors that is important. Detailed, complex information that is obligatorily hippocampally-dependent at acquisition remains so for its lifetime. However, all memories are semanticised as they age through the loss of detailed context-specific information and because generic cortically-represented information starts to dominate recall. Initially hippocampally dependent memories may appear to become independent of the hippocampus over time, but recall changes qualitatively. Multi-stage, lifelong post-acquisition memory processes produce semanticised re-representations of memories of differing specificity and complexity, that can serve different purposes. The model simulates hippocampal and cortical interactions in the acquisition and maintenance of episodic and semantic events, and behaves in accordance with my proposals. In particular, conceptualising episodic and semantic memory as representing points on a continuum of memory types appears viable. Support is also found for proposals on the relative importance of the hippocampus and cortex in the rapid acquisition of information and the acquisition of complex multi-model information; and the effect of existing knowledge on new learning. Furthermore, episodic and semantic events become differentially dependent on cortical and hippocampal components. Finally, as a memory ages, it is automatically semanticised and becomes cortically dependent.

In this thesis we investigate some of the issues involved in creating a reversible variant of the formal software development language B. We consider the effects of regarding computation as a potentially reversible process, yielding a number of new programming structures which we integrate into an implementation-level language RB0, a more expressive variant of B0, the current implementation-level language for B. Since reversibility simplifies garbage collection, in RB0 we make use of more abstract, set-based data types, normally available in B only at the specification level. Similarly, we propose extending the domain of abstract functions currently specifiable in B to allow them to become concrete functions, thereby furnishing B with a functional sub-language. We also investigate expanding the use of Lambda calculus from the abstract stage of B to the implementation. Unlike B0, RB0 will not disallow non-determinism, and can also specify what we call Prospective Value computations (which are described). The executable language implements all of these features. After introducing some preliminary concepts, we review the work leading to the rise of Reversible Computing as a possible answer to the growing problem of energy dissipation in modern processors. We describe the language RB0, and demonstrate the use of its features, introducing the companion language RB1 and its role in the process. We then introduce our execution platform, the Reversible Virtual Machine (RVM), and translate some of the examples developed earlier into RVM code. For the concrete functions, we provide a proposed syntax and translation schema to enable consistent translation to RVM, and introduce a postfix Lambda notation to link the RB0 specification to the RVM’s own postfix notation. We provide comprehensive translation schemas for those parts of RB0 which would be found in B operations; these will form the basis of an automated translation engine. In addition, we look at a denotational semantics for Bunch theory, which has proved useful in formalising the underlying concepts.

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Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 53). / In this thesis, we implement and test two algorithms for binary optimization and mixed integer optimization, respectively. We fine tune the parameters of these two algorithms and achieve satisfactory performance. We also compare our algorithms with CPLEX on large amount of fairly large-size instances. Based on the experimental results, our binary optimization algorithm delivers performance that is strictly better than CPLEX on instances with moderately dense constraint matrices, while for sparse instances, our algorithm delivers performance that is comparable to CPLEX. Our mixed integer optimization algorithm outperforms CPLEX most of the time when the constraint matrices are moderately dense, while for sparse instances, it yields results that are close to CPLEX, and the largest gap relative to the result given by CPLEX is around 5%. Our findings show that these two algorithms, especially the binary optimization algorithm, have practical promise in solving large, dense instances of both set covering and set packing problems. / by Jingting Zhou. / S.M.