Majorization arrow in quantum-algorithm design

We apply majorization theory to study the quantum algorithms known so far and find that there is a majorization principle underlying the way they operate. Grover's algorithm is a neat instance of this principle where majorization works step by step until the optimal target state is found. Extensions of this situation are also found in algorithms based in quantum adiabatic evolution and the family of quantum phase-estimation algorithms, including Shor's algorithm. We state that in quantum algorithms the time arrow is a majorization arrow.

Pseudo memory effects, majorization and entropy in quantum random walks

A quantum random walk on the integers exhibits pseudo memory effects, in that its probability distribution after N steps is determined by reshuffling the first N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy increase can be regarded as a consequence of the majorization ordering of successive distributions. The Lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general. Nevertheless, entropy can increase, and computer experiments show that it does so on average. Varying the stages at which the quantum coin system is traced out leads to new quantum walks, including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk.

On Majorization for Matrices

In this paper, we give several results for majorized matrices by using continuous convex function and Green function. We obtain mean value theorems for majorized matrices and also give corresponding Cauchy means, as well as prove that these means are monotonic. We prove positive semi-definiteness of matrices generated by differences deduced from majorized matrices which implies exponential convexity and log-convexity of these differences and also obtain Lypunov's and Dresher's type inequalities for these differences.

Weighted Euclidean biplots

We construct a weighted Euclidean distance that approximates any distance or dissimilarity measure between individuals that is based on a rectangular cases-by-variables data matrix. In contrast to regular multidimensional scaling methods for dissimilarity data, the method leads to biplots of individuals and variables while preserving all the good properties of dimension-reduction methods that are based on the singular-value decomposition. The main benefits are the decomposition of variance into components along principal axes, which provide the numerical diagnostics known as contributions, and the estimation of nonnegative weights for each variable. The idea is inspired by the distance functions used in correspondence analysis and in principal component analysis of standardized data, where the normalizations inherent in the distances can be considered as differential weighting of the variables. In weighted Euclidean biplots we allow these weights to be unknown parameters, which are estimated from the data to maximize the fit to the chosen distances or dissimilarities. These weights are estimated using a majorization algorithm. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing the matrix and displaying its rows and columns in biplots.

Blind Deconvolution via Lower-Bounded Logarithmic Image Priors

In this work we devise two novel algorithms for blind deconvolution based on a family of logarithmic image priors. In contrast to recent approaches, we consider a minimalistic formulation of the blind deconvolution problem where there are only two energy terms: a least-squares term for the data fidelity and an image prior based on a lower-bounded logarithm of the norm of the image gradients. We show that this energy formulation is sufficient to achieve the state of the art in blind deconvolution with a good margin over previous methods. Much of the performance is due to the chosen prior. On the one hand, this prior is very effective in favoring sparsity of the image gradients. On the other hand, this prior is non convex. Therefore, solutions that can deal effectively with local minima of the energy become necessary. We devise two iterative minimization algorithms that at each iteration solve convex problems: one obtained via the primal-dual approach and one via majorization-minimization. While the former is computationally efficient, the latter achieves state-of-the-art performance on a public dataset.

A Logarithmic Image Prior for Blind Deconvolution

Blind Deconvolution consists in the estimation of a sharp image and a blur kernel from an observed blurry image. Because the blur model admits several solutions it is necessary to devise an image prior that favors the true blur kernel and sharp image. Many successful image priors enforce the sparsity of the sharp image gradients. Ideally the L0 “norm” is the best choice for promoting sparsity, but because it is computationally intractable, some methods have used a logarithmic approximation. In this work we also study a logarithmic image prior. We show empirically how well the prior suits the blind deconvolution problem. Our analysis confirms experimentally the hypothesis that a prior should not necessarily model natural image statistics to correctly estimate the blur kernel. Furthermore, we show that a simple Maximum a Posteriori formulation is enough to achieve state of the art results. To minimize such formulation we devise two iterative minimization algorithms that cope with the non-convexity of the logarithmic prior: one obtained via the primal-dual approach and one via majorization-minimization.

Submodular Optimization and Data Processing

Thesis (Ph.D.)--University of Washington, 2016-06

Entanglement and boundary critical phenomena

We investigate boundary critical phenomena from a quantum-information perspective. Bipartite entanglement in the ground state of one-dimensional quantum systems is quantified using the Renyi entropy S-alpha, which includes the von Neumann entropy (alpha -> 1) and the single-copy entanglement (alpha ->infinity) as special cases. We identify the contribution of the boundaries to the Renyi entropy, and show that there is an entanglement loss along boundary renormalization group (RG) flows. This property, which is intimately related to the Affleck-Ludwig g theorem, is a consequence of majorization relations between the spectra of the reduced density matrix along the boundary RG flows. We also point out that the bulk contribution to the single-copy entanglement is half of that to the von Neumann entropy, whereas the boundary contribution is the same.