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.

In this chapter, we prove the Separation Theorem for convex sets and extract from it basic results on the geometry of closed convex sets, specifically (a) supporting hyperplanes, extreme points, and the (finite-dimensional) Krein--Milman Theorem, (b) recessive directions and recessive cone of a convex set, (c) the definition and basic properties of the dual cone, (d) the (finite-dimensional) Dubovitski--Milutin Lemma, (e) existence of bases and extreme rays of nontrivial closed pointed cones and their relation to extreme points of the cone’s base, (f)the (finite-dimensional) Krein--Milman Theorem in conic form, and (g) polarity.

The origin of decoherence of qubits is described by a simple example, and the two key methods to defeat decoherence, namely decoherence-free spaces and error-correcting codes are introduced.

>Mark in the following list the sets which are convex