After the conceptual comes the practical. While the study of quantum foundations is as old as quantum theory itself, the field of quantum computing is relatively new. So new in fact that large-scale, practical quantum computing is still not a reality. A typical ‘quantum computer’ takes many months to set up before performing such astounding tasks as factoring 6 into 3 × 2. Nonetheless, if those machines would exist, we know that we would gain amazing speed-ups in solving some hard (classical) computational problems, such as those involved in breaking a huge portion of cryptographic systems in use today.

Before we get into ‘quantum computing’, we should say a couple of things about ‘computing’. What is computing? Our answer by now probably won't come as such a shock: it's a process theory! Computation is indeed all about wiring the inputs and outputs of small processes together to make bigger processes. More specifically, a computation consists of a (finite) set of basic processes, which are wired together according to some (also finite) instructions, which we refer to as an algorithm or simply a program.

The only essential difference between classical and quantum computation is the contents of the basic processes. For classical computation, these operations consist of things like logical operations (e.g. XOR) or reading/writing locations in memory. For quantum computation, we can extend this with quantum processes and classical-quantum interactions such as measurements. So quantum computing is all about figuring out how to write new kinds of programs that exploit these new building blocks to build faster algorithms or accomplish new kinds of tasks that aren't possible classically.

The first quantum algorithms were ‘proofs of concept’, in the sense that they solved some problem much faster than a classical computer, but the kinds of problems they solved were not particularly interesting in their own right. However, this changed drastically with the advent of Grover's quantum search and Shor's factoring algorithms.

Although bankers on Wall Street can have pretty much whatever they want in whatever quantity they want, this chapter is for the rest of us cheapskates. Throughout this book, we have assumed that we, like quantum bankers, have had access to whatever processes we need in whatever quantity. For instance, in teleportation, we assumed we could obtain Bell states at will and do joint measurements on pairs of systems. For non-locality, we assumed we had lots of GHZ states around, so we could make enough measurements to see our assumptions about locality cave in. And for MBQC, we assumed that we had huge graph states around to implement universal quantum computation.

As you might expect, these things ain't cheap! Quantum processes involving multiple systems typically take some very special equipment and many hours to implement. So, when it comes to such resources, it behooves one to think about doing as much as possible with as little as possible. This is of course true not just for quantum states, but for any kind of resource: coal, oil, nuclear, wind, and solar energy; certain chemicals; or just a bit of affection.

What is important about all resources is what we can do with them. If we think of the benefits of resources (e.g. warm, comfy houses or secure communication) as being resources themselves, we see that pretty much all questions about resources boil down to the following two:

1. 1. Can a given resource be converted into some other resource?

2. 2. How much of resource X to do we need to obtain resource Y?

These are exactly the questions a resource theory aims to answer.

One place this idea of ‘conversion of resources’ appears very explicitly is in the study of chemical reactions, where one finds expressions like these:

Such an expression tells us that we can convert the stuff on the LHS to the stuff on the RHS. Even though the expression doesn't provide any details about how this conversion is actually done, it does provide us with two very useful pieces of information.

Glad you made it here! This book is about telling the story of quantum theory entirely in terms of pictures. Before we get into telling the story itself, it's worth saying a few words about how it came about. On the one hand, this is a very new story, in that it is closely tied to the past 10 years of research by us and our colleagues. On the other hand, one could say that it traces back some 80 years when the amazing John von Neumann denounced his own quantum formalism and embarked on a quest for something better. One could also say it began when Erwin Schrödinger addressed Albert Einstein's concerns about ‘spooky action at a distance’ by identifying the structure of composed systems (and in particular, their non-separability) as the beating heart of quantum theory.

From a complementary perspective, it traces back some 40 years when an undergraduate student named Roger Penrose noticed that pictures out-classed symbolic reasoning when working with the tensor calculus.

But 80 years ago the authors weren't around yet, at least not in human form, and 40 years ago there wasn't really that much of us either, so this preface will provide an egocentric take on the birth of this book. This also allows us to wholeheartedly acknowledge all of those without whom this book would never have existed (as well as some who nearly succeeded in killing it).

Things started out pretty badly for Bob, with a PhD in the 1990s on a then completely irrelevant topic of contextual ‘hidden variable representations’ of quantum theory – which recently have been diplomatically renamed to ontological models (Harrigan and Spekkens, 2010; Pusey et al., 2012). After a period of unemployment and a failed attempt to become a rock star, Bob ventured into the then even more irrelevant topic of von Neumann's quantum logic (Birkhoff and von Neumann, 1936) in the vicinity of the eccentric iconoclast Constantin Piron (1976).

It was there that category theory entered the picture, as well as serious considerations on the fundamental status of composition in quantum systems – something that went hand-in-hand with bringing quantum processes (rather than quantum states) to the forefront …