Here we discuss some of the interesting paradigm shifts that have been proposed for quantum computers: namely, using pseudo-pure states, cluster states, and non-deterministic gates.

Mark with Y those of the cases listed below where the linear form f⊤x separates the sets S and T

In this chapter, we demonstrate that (a) substituting the vector of eigenvalues of a symmetric n x n matrix into a convex permutation symmetric function of n real variables results in a convex function of the matrix, and (b) that if g is a convex function on the real axis, and G is the set of symmetric matrices of a given size with spectrum in the domain of g, then G is a convex set, and when X is a matrix from G, the trace of the matrix g(X), is a convex function of X; here g(X) is the matrix acting at a spectral subspace of X associated with eigenvalue v as multiplication by g(v); both these facts will be heavily used when speaking about cone-convexity is chapter 21.

In this chapter, we (a) outline the subject and the terminology of mathematical and convex programming, (b) introduce the Slater and relaxed Slater conditions and formulate the Convex Theorem on the Alternative -- the basis of Lagrange duality theory in convex programming, (c) introduce the notions of cone-convexity and of the convex programming problem in cone-constrained form, thus extending the standard mathematical programming setup of convex optimization, and (d) formulate and prove the Convex Theorem on the Alternative in cone-constrained form, justifying, as a byproduct, the standard Convex Theorem on the Alternative.

In this chapter, we derive the standard first- and second-order necessary/sufficient conditions for local optimality of a feasible solution to a (possibly nonconvex) mathematical programming problem. We conclude the chapter by illustrating these on the S-Lemma.

In this chapter, we (a) discuss the notion of lower semicontinuity of a function and demonstrate that functions with this property have closed epigraphs, (b) show that the pointwise supremum of a family of lower semicontinous functions is lower discontinuous, (c) demonstrate that a proper lower semiconscious convex function is the pointwise supremum of the affine minorants of the function, (d) introduce the notion of a subgradient and the subdifferential of a convex function at a point and demonstrate existence of subgradients at points from the relative interior of the function’s domain, (e) outline elementary rules of subdifferential calculus, and (f) establish basic properties of the directional derivatives of convex functions and the connection between directional derivatives and subdifferentials.

In this chapter, we extract from the results of Chapter 3 the basic theory of finite systems of linear inequalities - Farkas’ Lemmas, General Theorem on the Alternative, certificates for feasibility/infeasibility of polyhedral sets, and linear programming Duality Theorem.

After discussing the divorce of configuration and observable that is characteristic of the quantum description of reality, the reader is introduced to the awesome potential computational power that is afforded by quantum computation.

In this chapter, we (a) introduce the notion of Legendre transformation of a proper convex function, (b) establish basic properties of the Legendre transform, in particular, demonstrate that the transform of a proper lower semicontinuous convex function is itself a proper lower semicontinous convex function and that its Legendre transformation is the original function, (c) demonstrate that the set of minimizers of a proper lower semicontinuous convex function is the subdifferential, taken at the origin, of the function’s Legendre transform, and (d) derive the Young, Holder, and moment inequalities and discuss dual (a.k.a. conjugate) norms.

Stationary charges give rise to electric fields. Moving charges give rise to magnetic fields. In this chapter, we explore how this comes about, starting with currents in wires which give rise to a magnetic field wrapping the wire.

In this chapter, we rewrite the Maxwell equations yet again, this time in the language of actions and Lagrangians that we introduced in the first book in this series. This provides many new perspectives on electromagnetism. Among the pay-offs are a deeper understanding, via Noether’s theorem, of the energy and momentum carried by electromagnetism fields. This will also allow us to explore a number of deeper ideas, including superconductivity, the Higgs mechanism, and topological insulators.

In this chapter, we explore the basics of fluid mechanics. We will think about how to describe fluids and look at the kinds of things they can do.

Unusually, and a little defensively, the title of this chapter highlights what we won’t talk about, rather than what we will. Fluids have a property known as viscosity. This is an internal friction force acting within the fluid as diferent layers rub together. It is crucially important in many applications. In spite of its importance, we will start our journey into the world of fluids by ignoring viscosity altogether. Such flows are called inviscid. This will allow us to build intuition for the equations of fluid mechanics without the complications that viscosity brings. Moreover, the flows that we find in this section will not be wasted work. As we will see later, they give a good approximation to viscous flows in certain regimes where the more general equations reduce to those studied here.

The universe we live in is both strange and interesting. This strangeness comes about because, at the most fundamental level, the universe is governed by the laws of quantum mechanics. This is the most spectacularly accurate and powerful theory ever devised, one that has given us insights into many aspects of the world, from the structure of matter to the meaning of information. This textbook provides a comprehensive account of all things quantum. It starts by introducing the wavefunction and its interpretation as an ephemeral wave of complex probability, before delving into the mathematical formalism of quantum mechanics and exploring its diverse applications, from atomic physics and scattering, to quantum computing. Designed to be accessible, this volume is suitable for both students and researchers, beginning with the basics before progressing to more advanced topics.