Outside the immediate statistical applications, two notable implications of the Christoffel-Darboux kernel are sketched: on the effective semialgebraic approximation of nonsmooth functions and on the spectral analysis of Koopman's operator attached to some intricate dynamical systems.

A spectral characterization of the Christoffel function is provided via its associated moment matrix, and that of Dirac measures. A first (Hahn-Banach) extension of the Christoffel function to Lp spaces is characterized. A second extension is also provided and analyzed where squares of polynomials (hence positive everywhere) are replaced by larger convex cones of polynomials positive on the support of the underlying measure, in particular polynomials in a certain associated quadratic module.

A general overview of the book is detailed after a brief description of the main algorithm to compute and exploit Christoffel-Darboux from raw moment data.

Basic concepts and results of real and complex algebraic geometry enter into play when dealing with the estimation of measures supported by algebraic varieties. Some geometric constraints on the supporting variety are inherited from the very beginning of the modelization, such as a sphere for estimating an orientation in space. Well-adapted results of multivariate pluripotential theory play a crucial role here.

A brief introduction based on several classical examples to the correspondence between positive-definite kernels and Hilbert spaces of functions is tuned around the Christoffel-Darboux kernel and its relevance to moment problems.

This brief chapter discusses the minimum mathematical background required to understand the mathematical derivations in this text fully. A basic familiarity with matrices and vectors is assumed. The chapter introduces and reviews key properties of complex numbers, the Dirac notation with inner and outer products, the Kronecker product, unitary and Hermitian matrices, eigenvalues and eigenvectors, the matrix trace, and how to construct the Hermitian adjoint of matrix-vector expressions.

The basic infrastructure developed so far is sufficient for small-scale quantum algorithms. It is also a great learning tool. However, for complex algorithms with many more qubits and gates, this matrix-based infrastructure does not scale. This chapter improves the infrastructure to scale to problems with up to 30 qubits and tens of thousands of gates.

First, the chapter introduces an elegant circuit abstraction. A method to apply operators with linear complexity comes next, which is a significant improvement over the cubic or quadratic methods presented previously. Acceleration with C++ enables another 100x speedup. Finally, a sparse state representation is being discussed at length, which can be the best-performing implementation for many circuits.

This chapter introduces the fundamental concepts and rules of quantum computing. In parallel, it develops an initial, easy-to-understand codebase in Python for building and simulating small-scale quantum circuits and algorithms.

The chapter details single qubits, superposition, quantum states with many qubits, operators, including a sizable set of important single-qubit gates and controlled gates. The Bloch sphere and the quantum circuit notation are introduced. Entanglement follows, that fascinating “spooky action at a distance,” as Einstein called it. With this background, the chapter discusses maximally entangled Bell states, the no-cloning theorem, the noneffect of global phases, the partial trace and reduced density matrix, and uncomputation. The quantum postulates are discussed in a nonphilosophical way, leading to measurement and how to simulate it.

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.