Publications

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We present an approach to the simulation of quantum systems driven by classical stochastic processes that is based on the polynomial chaos expansion, a well-known technique in the field of uncertainty quantification. The polynomial chaos technique represents the density matrix as an expansion in orthogonal polynomials over the principle components of the stochastic process and yields a sparsely coupled hierarchy of linear differential equations. We provide practical heuristics for truncating this expansion based on results from time-dependent perturbation theory and demonstrate, via an experimentally relevant one-qubit numerical example, that our technique can be significantly more computationally efficient than Monte Carlo simulation. © 2013 American Physical Society.

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The expansion of a stochastic Liouville equation for the coupled evolution of a quantum system and an Ornstein-Uhlenbeck process into a hierarchy of coupled differential equations is a useful technique that simplifies the simulation of stochastically driven quantum systems. We expand the applicability of this technique by completely characterizing the class of diffusive Markov processes for which a useful hierarchy of equations can be derived. The expansion of this technique enables the examination of quantum systems driven by non-Gaussian stochastic processes with bounded range. We present an application of this extended technique by simulating Stark-tuned Förster resonance transfer in Rydberg atoms with nonperturbative position fluctuations. © 2012 American Physical Society.

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In previous work, we developed a Bayesian-based methodology to analyze the reliability of hierarchical systems. The output of the procedure is a statistical distribution of the reliability, thus allowing many questions to be answered. The principal advantage of the approach is that along with an estimate of the reliability, we also can provide statements of confidence in the results. The model is quite general in that it allows general representations of all of the distributions involved, it incorporates prior knowledge into the models, it allows errors in the 'engineered' nodes of a system to be determined by the data, and leads to the ability to determine optimal testing strategies. In this report, we provide the preliminary steps necessary to extend this approach to systems with feedback. Feedback is an essential component of 'complexity' and provides interesting challenges in modeling the time-dependent action of a feedback loop. We provide a mechanism for doing this and analyze a simple case. We then consider some extensions to more interesting examples with local control affecting the entire system. Finally, a discussion of the status of the research is also included.

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Complex systems are made up of multiple interdependent parts, and the behavior of the entire system cannot always be directly inferred from the behavior of the individual parts. They are nonlinear and system responses are not necessarily additive. Examples of complex systems include energy, cyber and telecommunication infrastructures, human and animal social structures, and biological structures such as cells. To meet the goals of infrastructure development, maintenance, and protection for cyber-related complex systems, novel modeling and simulation technology is needed. Sandia has shown success using M&S in the nuclear weapons (NW) program. However, complex systems represent a significant challenge and relative departure from the classical M&S exercises, and many of the scientific and mathematical M&S processes must be re-envisioned. Specifically, in the NW program, requirements and acceptable margins for performance, resilience, and security are well-defined and given quantitatively from the start. The Quantification of Margins and Uncertainties (QMU) process helps to assess whether or not these safety, reliability and performance requirements have been met after a system has been developed. In this sense, QMU is used as a sort of check that requirements have been met once the development process is completed. In contrast, performance requirements and margins may not have been defined a priori for many complex systems, (i.e. the Internet, electrical distribution grids, etc.), particularly not in quantitative terms. This project addresses this fundamental difference by investigating the use of QMU at the start of the design process for complex systems. Three major tasks were completed. First, the characteristics of the cyber infrastructure problem were collected and considered in the context of QMU-based tools. Second, UQ methodologies for the quantification of model discrepancies were considered in the context of statistical models of cyber activity. Third, Bayesian methods for optimal testing in the QMU framework were developed. This completion of this project represent an increased understanding of how to apply and use the QMU process as a means for improving model predictions of the behavior of complex systems. 4

A unitary quantum gate is the basic functioning element of a quantum computer. Summary of results: (1) Robustness of a general n-qubit gate = 1 - F {proportional_to} 2{sup n}; (2) Robustness of a universal gate with complete isolation of one-and two-qubit subgates = 1 - F {proportional_to} n; and (3) Robustness of a universal gate with small unwanted couplings between the qubits is unclear.

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The purpose of this project was to investigate the use of Bayesian methods for the estimation of the reliability of complex systems. The goals were to find methods for dealing with continuous data, rather than simple pass/fail data; to avoid assumptions of specific probability distributions, especially Gaussian, or normal, distributions; to compute not only an estimate of the reliability of the system, but also a measure of the confidence in that estimate; to develop procedures to address time-dependent or aging aspects in such systems, and to use these models and results to derive optimal testing strategies. The system is assumed to be a system of systems, i.e., a system with discrete components that are themselves systems. Furthermore, the system is 'engineered' in the sense that each node is designed to do something and that we have a mathematical description of that process. In the time-dependent case, the assumption is that we have a general, nonlinear, time-dependent function describing the process. The major results of the project are described in this report. In summary, we developed a sophisticated mathematical framework based on modern probability theory and Bayesian analysis. This framework encompasses all aspects of epistemic uncertainty and easily incorporates steady-state and time-dependent systems. Based on Markov chain, Monte Carlo methods, we devised a computational strategy for general probability density estimation in the steady-state case. This enabled us to compute a distribution of the reliability from which many questions, including confidence, could be addressed. We then extended this to the time domain and implemented procedures to estimate the reliability over time, including the use of the method to predict the reliability at a future time. Finally, we used certain aspects of Bayesian decision analysis to create a novel method for determining an optimal testing strategy, e.g., we can estimate the 'best' location to take the next test to minimize the risk of making a wrong decision about the fitness of a system. We conclude this report by proposing additional fruitful areas of research.