The exact and phaseless variants of auxiliary-field quantum Monte Carlo (AFQMC) have been shown to be capable of producing accurate ground-state energies for a wide variety of systems including those which exhibit substantial electron correlation effects. The first chapter of this thesis will provide an overview of the relevant electronic structure problem, and the phaseless AFQMC (ph-AFQMC) methodology. The computational cost of performing these calculations has to date been relatively high, impeding many important applications of these approaches. In Chapter 2 we present a correlated sampling methodology for AFQMC which relies on error cancellation to dramatically accelerate the calculation of energy differences of relevance to chemical transformations. In particular, we show that our correlated sampling-based ph-AFQMC approach is capable of calculating redox properties, deprotonation free energies, and hydrogen abstraction energies in an efficient manner without sacrificing accuracy. We validate the computational protocol by calculating the ionization potentials and electron affinities of the atoms contained in the G2 test set and then proceed to utilize a composite method, which treats fixed-geometry processes with correlated sampling-based AFQMC and relaxation energies via MP2, to compute the ionization potential, deprotonation free energy, and the O-H bond dissociation energy of methanol, all to within chemical accuracy. We show that the efficiency of correlated sampling relative to uncorrelated calculations increases with system and basis set size and that correlated sampling greatly reduces the required number of random walkers to achieve a target statistical error. This translates to reductions in wall-times by factors of 55, 25, and 24 for the ionization potential of the K atom, the deprotonation of methanol, and hydrogen abstraction from the O-H bond of methanol, respectively. In Chapter 3 we present an implementation of ph-AFQMC utilizing graphical processing units (GPUs). The AFQMC method is recast in terms of matrix operations which are spread across thousands of processing cores and are executed in batches using custom Compute Unified Device Architecture kernels and the hardware-optimized cuBLAS matrix library. Algorithmic advances include a batched Sherman-Morrison-Woodbury algorithm to quickly update matrix determinants and inverses, density-fitting of the two-electron integrals, an energy algorithm involving a high-dimensional precomputed tensor, and the use of single-precision floating point arithmetic. These strategies result in dramatic reductions in wall-times for both single- and multi-determinant trial wavefunctions. For typical calculations we find speed-ups of roughly two orders of magnitude using just a single GPU card. Furthermore, we achieve near-unity parallel efficiency using 8 GPU cards on a single node, and can reach moderate system sizes via a local memory-slicing approach. We illustrate the robustness of our implementation on hydrogen chains of increasing length, and through the calculation of all-electron ionization potentials of the first-row transition metal atoms. We compare long imaginary-time calculations utilizing a population control algorithm with our previously published correlated sampling approach, and show that the latter improves not only the efficiency but also the accuracy of the computed ionization potentials. Taken together, the GPU implementation combined with correlated sampling provides a compelling computational method that will broaden the application of ph-AFQMC to the description of realistic correlated electronic systems. In Chapter 4 the bond dissociation energies of a set of 44 3d transition metal-containing diatomics are computed with ph-AFQMC utilizing the correlated sampling technique. We investigate molecules with H, N, O, F, Cl, and S ligands, including those in the 3dMLBE20 database first compiled by Truhlar and co-workers with calculated and experimental values that have since been revised by various groups. In order to make a direct comparison of the accuracy of our ph-AFQMC calculations with previously published results from 10 DFT functionals, CCSD(T), and icMR-CCSD(T), we establish an objective selection protocol which utilizes the most recent experimental results except for a few cases with well-specified discrepancies. With the remaining set of 41 molecules, we find that ph-AFQMC gives robust agreement with experiment superior to that of all other methods, with a mean absolute error (MAE) of 1.4(4) kcal/mol and maximum error of 3(3) kcal/mol (parenthesis account for reported experimental uncertainties and the statistical errors of our ph-AFQMC calculations). In comparison, CCSD(T) and B97, the best performing DFT functional considered here, have MAEs of 2.8 and 3.7 kcal/mol, respectively, and maximum errors in excess of 17 kcal/mol (for the CoS diatomic). While a larger and more diverse data set would be required to demonstrate that ph-AFQMC is truly a benchmark method for transition metal systems, our results indicate that the method has tremendous potential, exhibiting unprecedented consistency and accuracy compared to other approximate quantum chemical approaches. The energy gap between the lowest-lying singlet and triplet states is an important quantity in chemical photocatalysis, with relevant applications ranging from triplet fusion in optical upconversion to the design of organic light-emitting devices. The ab initio prediction of singlet-triplet (ST) gaps is challenging due to the potentially biradical nature of the involved states, combined with the potentially large size of relevant molecules. In Chapter 5, we show that ph-AFQMC can accurately predict ST gaps for chemical systems with singlet states of highly biradical nature, including a set of 13 small molecules and the ortho-, meta-, and para- isomers of benzyne. With respect to gas-phase experiments, ph-AFQMC using CASSCF trial wavefunctions achieves a mean averaged error of ~1 kcal/mol. Furthermore, we find that in the context of a spin-projection technique, ph-AFQMC using unrestricted single-determinant trial wavefunctions, which can be readily obtained for even very large systems, produces equivalently high accuracy. We proceed to show that this scalable methodology is capable of yielding accurate ST gaps for all linear polyacenes for which experimental measurements exist, i.e. naphthalene, anthracene, tetracene, and pentacene. Our results suggest a protocol for selecting either unrestricted Hartree-Fock or Kohn-Sham orbitals for the single-determinant trial wavefunction, based on the extent of spin-contamination. These findings provide a reliable computational tool with which to investigate specific photochemical processes involving large molecules that may have substantial biradical character. We compute the ST gaps for a set of anthracene derivatives which are potential triplet-triplet annihilators for optical upconversion, and compare our ph-AFQMC predictions with those from DFT and CCSD(T) methods. We conclude with a discussion of ongoing projects, further methodological improvements on the horizon, and future applications of ph-AFQMC to chemical systems of interest in the fields of biology, drug-discovery, catalysis, and condensed matter physics.

One of the newer and rapidly developing approaches in quantum computing is based on "quantum walks," which are quantum processes on discrete space that evolve in either discrete or continuous time and are characterized by mixing of components at each step. The idea emerged in analogy with the classical random walks and stochastic techniques, but these unitary processes are very different even as they have intriguing similarities. This thesis is concerned with study of discrete-time quantum walks. The original motivation from classical Markov chains required for discrete-time quantum walks that one adds an auxiliary Hilbert space, unrelated to the one in which the system evolves, in order to be able to mix components in that space and then take the evolution steps accordingly (based on the state in that space). This additional, "coin," space is very often an internal degree of freedom like spin. We have introduced a general framework for construction of discrete-time quantum walks in a close analogy with the classical random walks with memory that is rather different from the standard "coin" approach. In this method there is no need to bring in a different degree of freedom, while the full state of the system is still described in the direct product of spaces (of states). The state can be thought of as an arrow pointing from the previous to the current site in the evolution, representing the one-step memory. The next step is then controlled by a single local operator assigned to each site in the space, acting quite like a scattering operator. This allows us to probe and solve some problems of interest that have not had successful approaches with "coined" walks. We construct and solve a walk on the binary tree, a structure of great interest but until our result without an explicit discrete time quantum walk, due to difficulties in managing coin spaces necessary in the standard approach. Beyond algorithmic interests, the model based on memory allows one to explore effects of history on the quantum evolution and the subtle emergence of classical features as "memory" is explicitly kept for additional steps. We construct and solve a walk with an additional correlation step, finding interesting new features. On the other hand, the fact that the evolution is driven entirely by a local operator, not involving additional spaces, enables us to choose the Fourier transform as an operator completely controlling the evolution. This in turn allows us to combine the quantum walk approach with Fourier transform based techniques, something decidedly not possible in classical computational physics. We are developing a formalism for building networks manageable by walks constructed with this framework, based on the surprising efficiency of our framework in discovering internals of a simple network that we so far solved. Finally, in line with our expectation that the field of quantum walks can take cues from the rich history of development of the classical stochastic techniques, we establish starting points for the work on non-Abelian quantum walks, with a particular quantum walk analog of the classical "card shuffling," the walk on the permutation group. In summary, this thesis presents a new framework for construction of discrete time quantum walks, employing and exploring memoried nature of unitary evolution. It is applied to fully solving the problems of: A walk on the binary tree and exploration of the quantum-to-classical transition with increased correlation length (history). It is then used for simple network discovery, and to lay the groundwork for analysis of complex networks, based on combined power of efficient exploration of the Hilbert space (as a walk mixing components) and Fourier transformation (since we can choose this for the evolution operator). We hope to establish this as a general technique as its power would be unmatched by any approaches available in the classical computing. We also looked at the promising and challenging prospect of walks on non-Abelian structures by setting up the problem of "quantum card shuffling," a quantum walk on the permutation group. Relation to other work is thoroughly discussed throughout, along with examination of the context of our work and overviews of our current and future work. / Graduation date: 2012

Also issued as a Ph.D. thesis in the Dept. of Electrical Engineering, 1970. / Bibliography: p. 168-173.

Also issued as a Sc.D. thesis in the Dept. of Electrical Engineering, 1969. / Bibliography: p.105.

Issued also as a Sc.D. thesis in the Dept. of Electrical Engineering, 1968. / Bibliography: p.103-104.

Based on a thesis in Electrical Engineering, 1966. / Bibliography: p.109-110. / Contract no. DA36-039-AMC-03200(E).

The discipline usually called `probability theory' can be seen as the theory which describes and sets standard norms to the way we reason about plausibility. From this point of view, this `plausibility theory' is a province of logic, and the following informal proportion subsists: plausibility theory is to the common notion of `plausibility', as deductive logic is to the common notion of `truth'. Some studies in plausibility theory are here offered. An alternative view and mathematical formalism for the problem of induction (the prediction of uncertain events from similar, certain ones) is presented. It is also shown how from plausibility theory one can derive a mathematical framework, based on convex geometry, for the description of the predictive properties of physical theories. Within this framework, problems like state assignment - for any physical theory - find simple and clear algorithms, numerical examples of which are given for three-level quantum systems. Plausibility theory also gives insights on various fashionable theorems, like Bell’s theorem, and various fashionable `paradoxes', like Gibbs' paradox. / QC 20100816

ABSTRACT “Theories of the Fantastic: Postmodernism, Game Theory, and Modern Physics” Karen Pike Degree: Doctor of Philosophy (2010) Centre for Comparative Literature University of Toronto This dissertation examines the fantastic mode of narrative as it appears in postmodern texts in a variety of media including literature, television, and film. By analyzing the kinds of changes which the fantastic mode has undergone in order to accommodate postmodern concerns, this project attempts to answer both how and why the fantastic has maintained its popularity and effectiveness. The first chapter seeks to define the fantastic mode by tracing the history of its definition from the early twentieth century up until the present. In doing so, it revisits the contributions of such analysts as Vax, Caillois, Todorov, and Freud. The second chapter discusses the changes to conventions demanded by postmodern discursive strategies, many of which include a back-and-forth movement between equally valid interpretations of the text. A discussion of Armin Ayren’s “Der Brandstifter,” a comparison of a recurring X-Files sub-plot to Bram Stoker’s Dracula, and an analysis of an intentionally self-reflexive episode of The X-Files demonstrate these changes. The third chapter introduces game theory as a way of understanding the back-and-forth movement typical of the fantastic mode. Hanns Heinz Ewers’s “Die Spinne” is used to illustrate the psychoanalytical aspect of this movement. The next chapter compares and contrasts three vampire films, The Addiction, Lair of the White Worm, and Nadja, in order to demonstrate how the degree to which this back-and-forth movement is present is an indicator of how successfully the fantastic effect emerges. The fifth chapter introduces modern physics as another mode for understanding the presence of the fantastic mode in the postmodern era. The analysis of House of Leaves in the final chapter illustrates how postmodern theory, game theory, and physics all work together to explain the fantastic’s effectiveness. This dissertation’s aim is to explain how and why a mode once defined as a specific nineteenth-century phenomenon keeps reinventing itself and re-emerging to continue to frighten and entertain us.

ABSTRACT “Theories of the Fantastic: Postmodernism, Game Theory, and Modern Physics” Karen Pike Degree: Doctor of Philosophy (2010) Centre for Comparative Literature University of Toronto This dissertation examines the fantastic mode of narrative as it appears in postmodern texts in a variety of media including literature, television, and film. By analyzing the kinds of changes which the fantastic mode has undergone in order to accommodate postmodern concerns, this project attempts to answer both how and why the fantastic has maintained its popularity and effectiveness. The first chapter seeks to define the fantastic mode by tracing the history of its definition from the early twentieth century up until the present. In doing so, it revisits the contributions of such analysts as Vax, Caillois, Todorov, and Freud. The second chapter discusses the changes to conventions demanded by postmodern discursive strategies, many of which include a back-and-forth movement between equally valid interpretations of the text. A discussion of Armin Ayren’s “Der Brandstifter,” a comparison of a recurring X-Files sub-plot to Bram Stoker’s Dracula, and an analysis of an intentionally self-reflexive episode of The X-Files demonstrate these changes. The third chapter introduces game theory as a way of understanding the back-and-forth movement typical of the fantastic mode. Hanns Heinz Ewers’s “Die Spinne” is used to illustrate the psychoanalytical aspect of this movement. The next chapter compares and contrasts three vampire films, The Addiction, Lair of the White Worm, and Nadja, in order to demonstrate how the degree to which this back-and-forth movement is present is an indicator of how successfully the fantastic effect emerges. The fifth chapter introduces modern physics as another mode for understanding the presence of the fantastic mode in the postmodern era. The analysis of House of Leaves in the final chapter illustrates how postmodern theory, game theory, and physics all work together to explain the fantastic’s effectiveness. This dissertation’s aim is to explain how and why a mode once defined as a specific nineteenth-century phenomenon keeps reinventing itself and re-emerging to continue to frighten and entertain us.

The primary study of this thesis is the experimental realization of the non-equilibrium dynamics of a quantum inverted pendulum as examined in the collective spin dynamics of a spin-1 Bose-Einstein condensate. In order to compare experimental results with the simulation past the low depletion limit, current simulation techniques needed to be extended to model atomic loss. These extensions show that traditional measurements of the system evolution (e.g. measuring the mean and standard deviation of the evolving quantity) were insufficient in capturing the quantum nature of the evolution. It became necessary to look at higher order moments and cumulants of the distributions in order to capture the quantum fluctuations. Extending the implications of the loss model further, it is possible that the system evolves in a way previously unpredicted. Spin-mixing from a hyperbolic fixed point in the phase space and low noise atom counting form the core of the experiment to measure the evolution of the distributions of the spin populations. The evolution of the system is also compared to its classical analogue, the momentum-shortened inverted pendulum. The other experimental study in this thesis is mapping the mean-field phase space. The mean-field phase space consists of different energy contours that are divided into both phase-winding trajectories and closed orbits. These two regions are divided by a separatrix whose orbit has infinite period. Coherent states can be created fairly accurately within the phase space and allowed to evolve freely. The nature of their subsequent evolution provides the shape of the phase space orbit at that initial condition. From this analysis a prediction of the nature of the entire phase space is possible.