Armed with the knowledge and infrastructure from the previous chapters, the first set of quantum algorithms is introduced. The algorithms in this chapter are typically shorter and require less preparation than those in later chapters. Additionally, the mathematical derivations are developed with great detail.

The chapter starts with the simplest possible algorithm: a quantum random number generator. This is followed by several gate equivalences, a classical full adder implemented with quantum gates, and the Swap Test to measure similarity between states. Two algorithms that utilize entanglement come next: quantum teleportation and superdense coding. After this, three so-called oracle algorithms are discussed,the Bernstein–Vazirani algorithm, Deutsch’s algorithm, and Deutsch–Jozsa’s algorithm. These are the first quantum algorithms that perform better than their classical counterparts. Oracle construction itself is discussed at great length.

This chapter details many of the fundamental quantum algorithms with full mathematical derivations and code. It discusses the quantum Fourier transform (QFT), arithmetic in the Fourier domain quantum phase estimation,Shor’s famous factorization algorithm and its quantum order-finding component, Grover’s search algorithm, amplitude amplification, and quantum counting, as well as quantum random walks.

From the field of quantum simulation, the chapter discusses the variational quantum eigensolver, measurement in the Pauli bases, the quantum approximate optimization algorithm (QAOA), the NP-complete graph Max-Cut problem, and the related Subset Sum problem. The chapter concludes with an in-depth discussion of the elegant Solovay–Kitaev algorithm for gate approximation.

This book is an introduction to quantum computing from the perspective of a classical programmer. Most concepts are explained with code, based on the insight that much of the complicated-looking math typically found in quantum computing may look quite simple in code. For many programmers, reading code is faster than reading complex math notation. Coding also allows experimentation, which helps with building intuition and understanding of the fundamental mechanisms of quantum computing.

This introductory chapter details the methodology used in this book and provides an overview of the major chapters. It suggests two alternative paths through the text, one with a focus on algorithms, the other focusing on infrastructure and simulation.

At this point, we understand the principles of quantum computing, the important foundational algorithms, and the basics of quantum error correction. We have developed a compact infrastructure for exploration and experimentation, but it is at the gate level.

Higher levels of abstraction are needed to scale to much larger programs. The chapter discusses several quantum programming languages, including their specific tooling, such as hierarchical program representations or entanglement analysis. General challenges for compilation are discussed, as well as compiler optimization techniques.

The chapter finishes with transpilation, a powerful compilation-based technique. It allows seamless porting of circuits to other frameworks, enabling the use of their advanced error models or distributed simulation capabilities. The underlying (simple) compiler technology would further enable implementation of several of the features found in programming languages, such as automatic uncomputation, entanglement analysis, and conditional blocks.

The term Beyond Classical is now the preferred term to describe computation that can be run efficiently on a quantum computer but would be intractable to run on a classical computer. A seminal paper by Google claimed to have reached this goal by computing a result in 200 seconds that a supercomputer would need 10,000 years to compute. Soon after publication, IBM claimed that the same computation could be done in just a few days on the Summit supercomputer. In this chapter, we analyze this disagreement. We discuss the proposition, implement and simulate the circuit, estimate simulation time for 53 qubits, and contrast Google’s claim against our implementation and IBM’s estimation result.

The Appendix contains additional interesting material. Specifically, it contains a detailed description of the implementation of the sparse simulation infrastructure, including failed and successful attempts at optimization.

This chapter discusses quantum noise and techniques for quantum error correction, a necessity for quantum computing. It discusses bit-flip errors, phase-flip errors, and their combination. The formalism of quantum operations is introduced, along with the operator-sum representation, and Krauss operators. With this, the chapter discusses the depolarization channel and imprecise gates, as well as (briefly) amplitude and phase damping. For error correction, repetition codes are introduced to motivate Shor’s 9-qubit error correction technique.

With this chapter starts the process of renormalization, how to assign in a consistent way a finite value to all these diverging integrals. The procedure itself is fully rigorous. It is the fact that the result of this procedure brings us information about the physical world which is largely an act of faith (which is however supported by experimental data to an astonishing degree of precision). We perform preliminary work, giving necessary conditions for rational functions to be integrable by comparing the degrees of the numerator and denominator, conditions which turn out to be sufficient for the rational functions of interest in computing the values of Feynman’s diagrams, a result known as Weinberg’s power counting theorem.

We make a short remark on the use of distribution theory to describe functions on regular surfaces.

We describe the BPHZ scheme of renormalization. The diverging integrals are tamed by replacing the integrand by a regularized version of it with better properties. Historically the scheme was recursive, removing in turn the divergences from the small diagrams to the larger ones, but now the regularization of the integrand is directly given as a sum of terms by the forest formula. In the end, the scheme can be seen as a very clever use of Taylor’s formula.

We answer some natural mathematical questions concerning representations. We develop the theory of induced representations for finite groups, which sheds considerable light on the structure of the induced representation of the Poincaré group studied in Chapter 8.

We provide the briefest introduction to Lagrangian and Hamiltonian Mechanics, and we explore several routes by which the physicists argue that the massive scalar field is a quantization of a natural classical field, the Klein-Gordon field.

The majority of our data concerning the particle world comes from scattering experiments, and the theoretical analysis of these is of fundamental importance. This analysis has two parts. First, we encode the properties of the scattering in an object called the S-matrix, whose computation is a main objective of the theory. Second, we relate the S-matrix to quantities that can actually be measured in our laboratory, the so-called cross-sections. We explain heuristically, through the analysis of situations of increasing complexity, what the S-matrix is, but we do not try yet to compute it. We then turn to the relation between the S-matrix and cross-sections, proving that indeed the S-matrix contains the information needed to predict the outcome of these experiments.