Related papers: Quantum permutations, Hadamard matrices, and the s…
In the paper is discussed complete probabilistic description of quantum systems with application to multiqubit quantum computations. In simplest case it is a set of probabilities of transitions to some fixed set of states. The probabilities…
Group algebras of permutations have proved highly useful in solving a number of problems in large N gauge theories. I review the use of permutations in classifying gauge invariants in one-matrix and multi-matrix models and computing their…
We explore the set of unitary matrices characterized by a given structure in the context of their applications in the field of Quantum Information. In the first part of the Thesis we focus on classification of special classes of unitary…
In this paper we disprove the Haagerup statement that all complex Hadamard matrices of order five are equivalent with the Fourier matrix $F_5$ by constructing circulant matrices that lead to new Hadamard matrices. An important item is the…
We briefly review the connection between the fuzzy field theories and matrix models and describe the main features of the models that appear. We summarize the different approaches to their analysis, some of the recent results and the…
This article is a short review on the relationship between convergent matrix integrals, formal matrix integrals, and combinatorics of maps. We briefly summarize results developed over the last 30 years, as well as more recent discoveries.…
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
This article discusses the essential difficulties in developing model-checking techniques for quantum systems that are never present in model checking classical systems. It further reviews some early researches on checking quantum…
We investigate linearity of amalgams of subgroups of algebraic groups along intersections with algebraic subgroups. In the process, we establish linearity of certain "doubles" of linear groups, and obtain new examples of finitely generated…
This is a short, self-contained expository survey, focused on algebraic and analytic aspects of quantum groups. Topics covered include the definition of ``quantum group,'' the Yang-Baxter equation, quantized universal enveloping algebras,…
Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found,…
A connection between nuclear symmetries other than those of an ellipsoidal nucleus and the properties of the implied rotational spectra are discussed. The discussion is focussed on a few examples of exotic shapes predicted recently by…
The relation between entanglement entropy and the computational difficulty of classically simulating Quantum Mechanics is briefly reviewed. Matrix product states are proven to provide an efficient representation of one-dimensional quantum…
Using the fact that any linear representation of a group can be embedded into permutations, we propose a constructive description of quantum behavior that provides, in particular, a natural explanation of the appearance of complex numbers…
Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application. Because of its relative maturity, computational…
We describe the notion of a quantum family of maps of a quantum space and that of a quantum commutant of such a family. Quantum commutants are quantum semigroups defined by a certain universal property. We give a few examples of these…
We give a basic overview of computational complexity, query complexity, and communication complexity, with quantum information incorporated into each of these scenarios. The aim is to provide simple but clear definitions, and to highlight…
The present survey aims at being a list of Conjectures and Problems in an area of model-theoretic algebra wide open for research, not a list of known results. To keep the text compact, it focuses on structures of finite Morley rank,…
Some very elementary ideas about quantum groups and quantum algebras are introduced and a few examples of their physical applications are mentioned.
Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a…