Do Ngoc Diep

Quantum Gauss-Jordan Elimination and Simulation of Accounting Principles on Quantum Computers

The paper is devoted to a new idea of simulation of accounting by quantum computing. We expose the actual accounting principles in a pure mathematics language. After that we simulated the accounting principles on quantum computers. We show that all arbitrary accounting actions are exhausted by the described basic actions. The main problem of accounting are reduced to some system of linear equations in the economic model of Leontief. In this simulation we use our constructed quantum Gau\ss-Jordan Elimination to solve the problem and the time of quantum computing is some square root order faster than the time in classical computing.The paper is devoted to a version of Quantum Gauss-Jordan Elimination and its applications. In the first part, we construct the Quantum Gauss-Jordan Elimination (QGJE) Algorithm and estimate the complexity of computation of Reduced Row Echelon Form (RREF) of N × N matrices. The main result asserts that QGJE has computation time is of order 2N/2. The second part is devoted to a new idea of simulation of accounting by quantum computing. We first expose the actual accounting principles in a pure mathematics language. Then, we simulate the accounting principles on quantum computers. We show that, all accounting actions are exhousted by the described basic actions. The main problems of accounting are reduced to some system of linear equations in the economic model of Leontief. In this simulation, we use our constructed Quantum Gauss-Jordan Elimination to solve the problems and the complexity of quantum computing is a square root order faster than the complexity in classical computing.

Noncommutative Chern Characters of Compact Lie Group C*-Algebras

In this paper we compute the K-theory (algebraic and topological) and entire periodic cyclic homology of compact Lie group C*-algebras, define Chern characters between them and show that the Chern characters in both topological and algebraic cases are isomorphisms.

Quantum Communication and Quantum Multivariate Polynomial Interpolation

The paper is devoted to the problem of multivariate polynomial interpolation and its application to quantum secret sharing. We show that using quantum Fourier transform one can produce the protocol for quantum secret sharing distribution.

Category of Noncommutative CW-Complexes. II

We introduce in this paper the notion of noncommutative Serre fibration (shortly, NCSF) and show that up to homotopy, every morphism between NCCW-complexes is some noncommutative Serre fibration. We then associate a six-term exact sequence with the periodic cyclic homology and for K-theory of an arbitrary noncommutative Serre fibration. We also show how to use this technique to compute K-groups and cyclic theory groups of some noncommutative quotients. This paper is a follow-up of ideas in Diep (K-Theory Archiv 153, 2007, Vietnam J. Math. 38:363–371, 2010).

A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry group . Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundles with connection over the space dual to the Lie algebra of the symmetry group . After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry group . Use the electric-magnetic duality to pass to the Langlands dual Lie group . Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra 𝔤. Use the construction of Fock space reprsentations to have representations of such affine loop algebra. And finally, we have the automorphic representations of the corresponding Langlands-dual Lie groups .

Riemann-Roch Theorem and Index Theorem in Non-commutative Geometry ∗

In this lecture we review apprearance of the Riemann-Roch Theorem in classical function theory, Algebraic topology, in theory of pseudo-differential operators and finally in noncommutative geometry. We show also it usefulness in many problem of quantization and harmonic analysis on Lie groups.