Related papers: Quantum $K$-theory and $q$-Difference equations
This is a position paper written as an introduction to the special volume on quantum algorithms I edited for the journal Mathematical Structures in Computer Science (Volume 20 - Special Issue 06 (Quantum Algorithms), 2010).
In these lectures, we discuss two approaches to studying orbit spaces of algebraic Lie groups. Due to algebraic approach orbit space, or quotient, is an algebraic manifold, while from the differential viewpoint a quotient is a differential…
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from…
These notes give a brief introduction to differential Harnack inequalities and summarise the main results of the mini-course ``Li-Yau and Harnack estimates for nonlocal diffusion problems'', presented by the author at the Seasonal School on…
We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time.…
We give an overview of linear optics quantum computing, focusing on the results from the original KLM paper. First we give a brief summary of the advances made with optics for quantum computation prior to KLM. We next discuss the KLM linear…
"Ever since the advent of modern quantum mechanics in the late 1920's, the idea has been prevalent that the classical laws of probability cease, in some sense, to be valid in the new theory. [...] The primary object of this presentation is…
Although deep learning (DL) has already become a state-of-the-art technology for various data processing tasks, data security and computational overload problems often arise due to their high data and computational power dependency. To…
In this introductory review, we focus on applications of quantum computation to problems of interest in physics and chemistry. We describe quantum simulation algorithms that have been developed for electronic-structure problems,…
This paper gives an overview from the perspective of Lie group theory of some of the recent advances in the rapidly expanding research area of quantum entanglement. This paper is a written version of the last of eight one hour lectures…
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by being based on two…
We will show for undergraduate and graduate students of physics that Quantum Mechanics is an incomplete and non-local theory. The problem of non-locality is discussed by analyzing the Bell's theorem where are considered correlations between…
We develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main…
This is a set of lecture notes for a graduate-level course on quantum algorithms, with an emphasis on quantum optimization algorithms. It is developed for applied mathematicians and engineers, and requires no previous background in quantum…
This survey paper is an expanded version of lectures given at the Clay Mathematics Academy ; see http://www.claymath.org/programs/outreach/academy/colloquium2005.php These lectures were intended to very young (and motivated) college…
Lecture notes with a brief introduction to Quantum field theory and the Standard Model are presented. The lectures were given at the 2017 European School of High-Energy Physics. The main features, the present status, and problems of the…
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…
In loop quantum cosmology, a difference equation for the wave function describes the evolution of a universe model. This is different from the differential equations that arise in Wheeler-DeWitt quantizations, and some aspects of general…
Quantum theory departs from classical probabilistic theories in foundational ways. These departures--termed quantumness here--power quantum information and computation. This thesis charts the role of discrete structures in assessing…
A causal, non-Hermitian, renormalizable, local, unitary and Lorentz convariant formulation of Quantum Theory (QT) (= Quantum Mechanics (QM) and Quantum Field Theory (QFT)) is developed which is free of formalistic problems we face in the…