Related papers: Quantum permutations, Hadamard matrices, and the s…
Very recently the most general ensemble of qubits are identified using the notion of linearity; any of these qubits gets accepted by a Hadamard gate to generate the equal superposition of the qubit and its orthogonal. Towards more…
Quantum computing provides a powerful framework for tackling computational problems that are classically intractable. The goal of this paper is to explore the use of quantum computers for solving relevant problems in systems and control…
Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling Abelian groups and constructions of complex Hadamard…
Random matrix models have been extensively studied in mathematical physics and have proven useful in combinatorics. In this review paper we introduce a generalization of these models to a class of tensor models. As the topology and…
We obtain the most general ensemble of qubits, for which it is possible to design a universal Hadamard gate. These states when geometrically represented on the Bloch sphere, give a new trajectory. We further consider some Hadamard `type' of…
Convenient parameterizations of matrices in terms of vectors transform (certain classes of) matrix equations into covariant (hence rotation-invariant) vector equations. Certain recently introduced such parameterizations are tersely…
We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the…
The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of…
The driving force in the pursuit for quantum computation is the exciting possibility that quantum algorithms can be more efficient than their classical analogues. Research on the subject has unraveled several aspects of how that can happen.…
Hadamard matrices are square $n\times n$ matrices whose entries are ones and minus ones and whose rows are orthogonal to each other with respect to the standard scalar product in $\Bbb R^n$. Each Hadamard matrix can be transformed to a…
Some formulas and speculations are presented relative to integrable systems and quantum mechanics.
The general linear group acts on $m$-tuples of $N\times N$ matrices by simultaneous conjugation. Quantum deformations of the corresponding rings of invariants and the so-called trace rings are investigated.
In this paper, we give several matrix representations for the Horadam quaternions. We derive several identities related to these quaternions by using the matrix method. Since quaternion multiplication is not commutative, some of our results…
A partial Hadamard matrix $H\in M_{M\times N}(\mathbb C)$ is called of "classical type" if the associated quantum semigroup $G\subset\widetilde{S}_M^+$ is classical. In combinatorial terms, if $H_1,\ldots,H_M\in\mathbb T^N$ are the rows of…
In this survey, we discuss some basic problems concerning random matrices with discrete distributions. Several new results, tools and conjectures will be presented.
We consider two different types of deformations for the linear group $ GL(n)$ which correspond to using of a general diagonal R-matrix. Relations between braided and quantum deformed algebras and their coactions on a quantum plane are…
In this survey article we give basic introduction to the theory of quantum families of maps. We begin with a general look at non-commutative (or "quantum") topology. Then we formulate all our results in this language. Existence of quantum…
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…
This is a survey on the transitive quantum groups $G\subset S_N^+$, and on the flat matrix models $\pi:C(G)\to M_N(C(X))$ for the corresponding Hopf algebras. We review the known results on the subject, with a number of improvements,…
Few, if any, applications of quantum technology are as widely known as the quantum simulation of quantum matter. Consequently, many interesting questions have been sparked at the intersection of condensed matter, quantum chemistry, and…