相关论文: An introduction to quantum groups and non-commutat…
These notes partly touch the topic of the talk given by the author at the XXXVIII Workshop on Geometric Methods in Physics, hold in June-July 2019 in Bia\l{}owie\.{z}a, Poland. They consist of a short and self-contained introduction to the…
A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential…
Condensed account of the Lectures delivered at the Meeting on {\it Noncommutative Geometry in Field and String Theory}, Corfu, September 18 - 20, 2005.
Quantum groups lead to an algebraic structure that can be realized on quantum spaces. These are noncommutative spaces that inherit a well defined mathematical structure from the quantum group symmetry. In turn such quantum spaces can be…
We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by…
Quantum algorithms for factoring and discrete logarithm have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden…
This is an introduction for nonspecialists to the noncommutative geometric approach to Planck scale physics coming out of quantum groups. The canonical role of the `Planck scale quantum group' $C[x]\bicross C[p]$ and its observable-state…
An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand-Naimark and Serre-Swan equivalences and thus allows one to represent…
This is a very brief introduction to quantum computing and quantum information theory, primarily aimed at geometers. Beyond basic definitions and examples, I emphasize aspects of interest to geometers, especially connections with asymptotic…
These Lectures are based on a course on noncommutative geometry given by the author in 2003 at the University of Chicago. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild…
We offer an alternative to the conventional network formulation of quantum computing. We advance the analog approach to quantum logic gate/circuit construction. As an illustration, we consider the spatially extended NOT gate as the first…
We carry out a generalization of quantum group co-representations in order to encode in this structure those cases where non-commutativity between endomorphism matrix entries and quantum space coordinates happens.
In this paper, we construct a covariant differential calculus on quantum plane with two-parametric quantum group as a symmetry group. The two cases $d^2=0$ and $d^3=0$ are completly established. We also construct differential calculi $n=2$…
This article gives an elementary introduction to quantum computing. It is a draft for a book chapter of the "Handbook of Nature-Inspired and Innovative Computing", Eds. A. Zomaya, G.J. Milburn, J. Dongarra, D. Bader, R. Brent, M.…
Notes of a 8h course given at the University of G\"oteborg during an Erasmus exchange visit, June 11-15, 2018. It is intended for PhD and graduate students familiar with $C^*$-algebras but not specializing in quantum groups. The proofs, if…
We define and study noncommutative generalizations of submanifolds and quotient manifolds, for the derivation-based differential calculus introduced by M.~Dubois-Violette and P.~Michor. We give examples to illustrate these definitions.
These are the expanded notes of a course given at the Summer school "Geometric, topological and algebraic methods for quantum field theory" held at Villa de Leyva, Colombia in July 2015. We first give an introduction to non-commutative…
We first give a pedagogical introduction to the differential calculus on q-groups and analize the relation between differential calculus and q-Lie algebra. Equivalent definitions of bicovariant differential calculus are studied and their…
Based on results for real deformation parameter q we introduce a compact non- commutative structure covariant under the quantum group SOq(3) for q being a root of unity. To match the algebra of the q-deformed operators with necesarry…
These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.