arXiv subjects

Kevin Schultz

Publications and source records attributed to Kevin Schultz.

21 records · Page 2Linked to original sources

arXiv2020

Graph Signal Processing of Indefinite and Complex Graphs using Directed Variation

In the field of graph signal processing (GSP), directed graphs present a particular challenge for the "standard approaches" of GSP to due to their asymmetric nature. The presence of negative- or complex-weight directed edges, a graphical structure used in fields such as neuroscience, critical infrastructure, and robot coordination, further complicates the issue. Recent results generalized the total variation of a graph signal to that of directed variation as a motivating principle for developing a graphical Fourier transform (GFT). Here, we extend these techniques to concepts of signal variation appropriate for indefinite and complex-valued graphs and use them to define a GFT for these classes of graph. Simulation results on random graphs are presented, as well as a case study of a portion of the fruit fly connectome.

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arXiv2019

Exponential Families for Bayesian Quantum Process Tomography

A Bayesian approach to quantum process tomography has yet to be fully developed due to the lack of appropriate probability distributions on the space of quantum channels. Here, by associating the Choi matrix form of a completely positive, trace preserving (CPTP) map with a particular space of matrices with orthonormal columns, called a Stiefel manifold, we present two parametric probability distributions on the space of CPTP maps that enable Bayesian analysis of process tomography. The first is a probability distribution that has an average Choi matrix as a sufficient statistic. The second is a distribution resulting from binomial likelihood data that enables a simple connection to data gathered through process tomography experiments. To our knowledge these are the first examples of continuous, non-unitary random CPTP maps, that capture meaningful prior information for use in Bayesian estimation. We show how these distributions can be used for point estimation using either maximum a posteriori estimates or expected a posteriori estimates, as well as full Bayesian tomography resulting in posterior credibility intervals. This approach will enable the full power of Bayesian analysis in all forms of quantum characterization, verification, and validation.

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arXiv2016

Orientation Statistics and Quantum Information

Motivated by the engineering applications of uncertainty quantification, in this work we draw connections between the notions of random quantum states and operations in quantum information with probability distributions commonly encountered in the field of orientation statistics. This approach identifies natural probability distributions that can be used in the analysis, simulation, and inference of quantum information systems. The theory of exponential families on Stiefel manifolds provides the appropriate generalization to the classical case, and fortunately there are many existing techniques for inference and sampling that exist for these distributions. Furthermore, this viewpoint motivates a number of additional questions into the convex geometry of quantum operations relative to both the differential geometry of Stiefel manifolds as well as the information geometry of exponential families defined upon them. In particular, we draw on results from convex geometry to characterize which quantum operations can be represented as the average of a random quantum operation.

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