In this chapter, we (a) present the notion of a polyhedral representation and illustrate its importance, (b) demonstrate via Fourier--Motzkin eliminaton that every polyhedrally representable set is polyhedral, and (c) outline the calculus of polyhedral representations. As an immediate application, we demonstrate that a bounded and feasible LP problem is solvable.

In this chapter, we (a) present epigraph characterization of cone-convexity, (b) introduce cone-monotonicity, and describe differential criteria of cone-convexity and cone-monotonicity, (c) present instructive examples of cone-convex and cone-monotone functions, (d) outline basic operations preserving cone-convexity and cone-monotonicity. Taken together, (b)--(d) provide simple and powerful tools allowing one to detect and utilize cone-convexity and cone-monotonicity.

In this chapter, we (a) present an algebraic characterization of extreme points of polyhedral sets and extreme rays of polyhedral cones, (b) describe extreme points of several important polyhedral sets, including the Birkhoff--von-Neumann Theorem on extreme points of the polytope of doubly stochastic matrices, (c) establish the theorem on the structure of polyhedral sets stating that nonempty polyhedral sets are exactly the sets representable as sums of convex hulls of nonempty finite sets and conic hulls of finite sets, and vice versa, (d) extract from the latter theorem basic descriptive results of linear programming theory, and (e) present and justify the Majorization Principle.

In this chapter, we present preliminaries on convex functions -- definitions via convexity inequality and via the convexity of the epigraph, basic examples, Jensen’s inequality, convexity of sublevel sets; we introduce the notion of the domain of a convex function and its representation as a function taking values in the extended real axis and introduce the concept of a proper convex function.

Deducing the quantum state of your device is essential for diagnosing and perfecting it, and the methods needed for this are introduced in this chapter. We also extend the discussion to methods used to validate noisy, intermediate-scale quantum computers when they grow too large for tomography to be used.

The generic properties of physical qubits are discussed in detail: in particular the need for an energy gap to ensure cooling and its implications for the size of devices. The basic notions of controlling qubits by external forces shows us how single-qubit gates are implemented.

In this chapter, we present and illustrate Caratheodory’s Theorem (in plain and conic forms), Radon’s Theorem, and Helly’s Theorem (for finite and for infinite families of convex sets).

In this chapter, we (a) present the definition and game theory interpretation of saddle points, (b) describe primal and dual optimization problems induced by an antagonistic game, (c) provide a characterization of saddle points in terms of optimal solutions and optimal values of primal and dual problems induced by the game, and (d) formulate and prove he Minimax Lemma and the Sion--Kakutani Theorem on existence of saddle points in convex--concave antagonistic games.

Several other technologies under development to exploit quantum power are discussed in this chapter. You will learn about quantum key distribution; improving measurements of phase shifts is used as an example to demonstrate the power of entanglement in beating the standard quantum limit. How the latter is used to improve detection of objects is also discussed. Finally, modelling complicated quantum systems by designing simpler and easier to control systems, represented by quantum circuits, simplifies the studying of such systems, allowing us to gain better insight into their physics and to make better predictions about them.

In this chapter, we (a) demonstrate that every local minimizer of a convex function is its global minimizer, (b) show that the Fermat rule provides a necessary and sufficient condition for an interior point of the domain of a convex function to be a global minimizer of the function, provided the function is differentiable at the point, (c) introduce radial and normal cones and express in terms of these cones the necessary and sufficient condition for a point to be the minimizer of a convex function whenever the function is differentiable at the point, (d) introduce the symmetry principle and (e) provide basic information on maximizers of convex functions.

Which of the following functions are convex on the indicated domains

Quantum computing technology was born in the 1970s and 1980s when a handful of visionary thinkers such as Paul Benioff, Richard Feynman, and David Deutsch first speculated about how the precepts of quantum mechanics might impact computer science. In 1984 Gilles Brassard, a computer scientist and cryptographer, and Charles Bennett, a specialist in physics and information theory, devised a practical application for quantum mechanics in the field of secure communication.

Here we build the skills needed to master how a quantum computer can factor very large numbers much more efficiently than a classical computer; i.e., it is a chapter dedicated to Shor’s algorithm. The Fourier transform, and its quantum analogue are introduced and applied to period finding. These are then applied to show how the problem of factoring large numbers amounts to finding the period of a modular exponential function. Moreover, the consequences of such a capability on the everyday security in (internet) communications using RSA encryption is also discussed.

In this chapter, we (a) present the definition and basic examples (linear and affine subspace, polyhedron, norm ball, simplex, cone) of convex sets, (b) introduce and study the notions of convex combination and convex hull, (c) outline basic convexity-preserving operations with sets, (d) establish nonemptiness of the relative interior of a nonempty convex set and the fact that the relative interior of a convex sets is dense in its closure, and (e) introduce the notions of conic and perspective transforms of a convex set.

In this chapter, we (a) introduce the Lagrange function of an inequality-constrained mathematical programming problem, (b) formulate and prove convex programming Lagrange Duality Theorem, and (c) establish the connection between Lagrange duality and saddle points of the Lagrange function.