The field of quantum computing is developing at a rapid pace, and one can expect paradigm-shifting advances in coming years. The goal of this chapter is for the reader to understand fully the language and basic concepts of quantum information science needed to engage in research and development in this very exciting field in the future. We will apply the mathematical machinery we have acquired so far to develop the quantum counterparts to the classical notions of bits, logic gates, circuits, and algorithms. We will also review some of the promising examples of quantum hardware for physically realizing quantum information processing.

A regrettable amount of mathematical machinery goes into a good understanding of quantum mechanics. This could be avoided if a good intuitive understanding of many quantum systems was possible, but as intuition is generally derived from daily experience (which is governed by classical laws), we cannot expect this to be the case in general. Here, we present an in-depth introduction to the mathematical foundations of quantum mechanics, accompanied, wherever appropriate, by detailed explanations of relevant quantum concepts such as superposition, wavefunction collapse, and the uncertainty principle. As an additional benefit, the language developed in this chapter will be especially useful for describing quantum information science in .

A clause consists of a subject and a predicate. The subject is always a noun or noun phrase, or a verbal element used as a noun. The predicate, in a verbal clause, consists of the verb and its complements (ch. 12); in a copular clause, it consists of the copula (expressed or unexpressed) and a noun/noun phrase, adjective/adjective phrase, or a prepositional phrase (§373). These are the core elements of a clause. Existential clauses with איתי (§380) and presentative constructions (§382), however, do not have subjects.

Now that we have developed some familiarity with structures, we can turn our attention to implication. In the propositional logic setting, we defined two ways to say that a set $\Gamma \subseteq {\text{Form}}_P$ implies a formula $\phi \in {\text{Form}}_P$. In that case, the semantic approach involved examining truth assignments, and the syntactic approach relied on the development of a formal proof system. We then established, in the Soundness and Completeness Theorems, that these two concepts coincided. By jumping between the two approaches, we found the Compactness Theorem as a nice corollary.

Chapter 8 focused on describing first-order axiomatic set theory, and then showing how to embed mathematics within that theory. But is there anything more that the set-theoretic perspective provides to the mathematical tool kit beyond a unifying foundation and cute diagonal arguments?

One of the successful results of such a program is the ability to study mathematical language and reasoning using mathematics itself. For example, we will eventually give a precise mathematical definition of a formal proof, and to avoid confusion with our current intuitive understanding of what a proof is, we will call these objects deductions. One can think of our eventual definition of a deduction as analogous to the precise mathematical definition of continuity, which replaces the fuzzy “a graph that can be drawn without lifting your pencil.” Once we have codified the notion in this way, we will have turned deductions into precise mathematical objects, allowing us to prove mathematical theorems about deductions using normal mathematical reasoning. For example, we will open up the possibility of proving that there is no deduction of certain mathematical statements.

One of the major triumphs of early twentieth-century logic was the formulation of several (equivalent) precise definitions for what it means to say that a function $f:\mathbb{N}\to \mathbb{N}$ is computable. In our current age, many people have an intuitive sense of this concept through experience with computer programming. However, it is challenging to turn such intuition into a concise formal mathematical treatment that is susceptible to a rigorous mathematical analysis.

Set theory originated in an attempt to understand and somehow classify small, or negligible, sets of real numbers. Cantor’s early explorations into the realm of the transfinite were motivated by a desire to understand the points of convergence of trigonometric series. The basic ideas quickly became a fundamental part of analysis, in addition to permeating many other areas of mathematics. Since then, set theory has become a way to unify mathematical practice, and the way in which mathematicians grapple with the infinite in all areas of mathematics.

In many areas of mathematics (like partial orderings, groups, or graphs), we write down some axioms and immediately have several different models of these axioms in mind. In the setting of first-order logic, this corresponds to writing down a set Σ of sentences in a language and looking at the elementary class $\text{Mod}(\sum )$. Since $\text{Mod(Σ)}=\text{Mod(Cn(Σ))}$ by Proposition 6.5.3, and Cn(Σ) is a theory by Proposition 6.5.4, we can view this situation as looking at the (elementary) class of models of a theory.

This case study incorporates the different plans discussed in Chapter 7 and the diffusion of an environmental innovation. This is a case history demonstrating how change agents working with the EPA developed and implemented Integrated Pest Management (IPM) in schools in 41 states in 18 years and how, for more than a decade, School IPM became a national initiative. The case study first appeared in A Worm in the Teacher’s Apple: Protecting America’s School Children from Pests and Pesticides (Lame 2005)