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Unitary k-designs are probabilistic ensembles of unitary matrices whose first k statistical moments match that of the full unitary group endowed with the Haar measure. In prior work, we showed that the automorphism group of classical Z 4 -linear Kerdock codes maps to a unitary 2-design, which established a new classical-quantum connection via graph states. In this paper, we construct a Markov process that mixes this Kerdock 2-design with symplectic transvections, and show that this process produces an ϵ -approximate unitary 3-design. We construct a graph whose vertices are Pauli matrices, and two vertices are connected by directed edges if and only if they commute. A unitary ensemble that is transitive on vertices, edges, and non-edges of this Pauli graph is an exact 3-design, and the stationary distribution of our process possesses this property. With respect to the symmetries of Kerdock codes, the Pauli graph has two types of edges; the Kerdock 2-design mixes edges of the same type, and the transvections mix the types. More precisely, on m qubits, the process samples O (log (N 5 / ϵ)) random transvections, where N = 2 m , followed by a random Kerdock 2-design element and a random Pauli matrix. Hence, the simplicity of the protocol might make it attractive for several applications. From a hardware perspective, 2-qubit transvections exactly map to the Mølmer–Sørensen gates that form the native 2-qubit operations for trapped-ion quantum computers. Thus, it might be possible to extend our work to construct an approximate 3-design that only involves such 2-qubit transvections. [ABSTRACT FROM AUTHOR]

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame can be regarded as a frame-like collection of subspaces in a Hilbert space, and thereby generalizes the concept of a frame for signal representation. However, when the signal and/or subspace dimensions are large, the decomposition of the signal into its fusion frame measurements through subspace projections typically requires a large number of additions and multiplications, and this makes the decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we introduce the notion of a sparse fusion frame, that is, a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a 'uniform basis' over all subspaces, thereby enabling low-complexity fusion frame decompositions. We study the existence and construction of sparse fusion frames, but our focus is on developing simple algorithmic constructions that can easily be adopted in practice to produce sparse fusion frames with desired (given) operators. By a desired (or given) operator we simply mean one that has a desired (or given) set of eigenvalues for the fusion frame operator. We start by presenting a complete characterization of Parseval fusion frames in terms of the existence of special isometries defined on an encompassing Hilbert space. We then introduce two general methodologies to generate new fusion frames from existing ones, namely the Spatial Complement Method and the Naimark Complement Method, and analyze the relationship between the parameters of the original and the new fusion frame. We proceed by establishing existence conditions for 2-sparse fusion frames for any given fusion frame operator, for which the eigenvalues are greater than or equal to two. We then provide an easily implementable algorithm for computing such 2-sparse fusion frames. [ABSTRACT FROM AUTHOR]

Pump-probe microscopy is an emerging technique that provides detailed chemical information of absorbers with sub-micrometer spatial resolution. Recent work has shown that the pump-probe signals from melanin in human skin cancers correlate well with clinical concern, but it has been difficult to infer the molecular origins of these differences. Here we develop a mathematical framework to describe the pump-probe dynamics of melanin in human pigmented tissue samples, which treats the ensemble of individual chromophores that make up melanin as Gaussian absorbers with bandwidth related via Frenkel excitons. Thus, observed signals result from an interplay between the spectral bandwidths of the individual underlying chromophores and spectral proximity of the pump and probe wavelengths. The model is tested using a dual-wavelength pump-probe approach and a novel signal processing method based on gnomonic projections. Results show signals can be described by a single linear transition path with different rates of progress for different individual pump-probe wavelength pairs. Moreover, the combined dual-wavelength data shows a nonlinear transition that supports our mathematical framework and the excitonic model to describe the optical properties of melanin. The novel gnomonic projection analysis can also be an attractive generic tool for analyzing mixing paths in biomolecular and analytical chemistry. [ABSTRACT FROM AUTHOR]

Presents an obituary of the inventor and mathematician Claude Shannon. His central role in starting the digital revolution; Influence of his mathematical theory of communication in the field of information theory; Educational background and career highlights; His interest in circuit design; Examples of his inventions; Opinion that he is one of the major figures of twentieth century science.