$|v|sin\theta {\widehat{\mathbf{\text{u}}}}_{\mathbf{\perp }}.\widehat{\mathbf{\text{u}}}=0$ (since ${\widehat{\mathbf{\text{u}}}}_{\mathbf{\perp }}.\widehat{\mathbf{\text{u}}}=0$). Therefore, the projection of $\widehat{\mathbf{\text{v}}}$ on $\widehat{\mathbf{\text{u}}}$ subtracted from $\widehat{\mathbf{\text{v}}}$ gives a vector that is perpendicular to $\widehat{\mathbf{\text{v}}}$

We introduce the reader to the physics underlying four key qubit technologies: photons, spins, ions, and superconducting circuits, and their pros and cons are discussed.

The key issue of two-qubit gates is discussed in this chapter: there are two basic approaches: direct interaction (which is easy but short-ranged) and using a quantum data bus, which is the key ingredient of the Cirac-Zoller gate.

Using linear algebra, the mathematical techniques needed for describing and manipulating qubits are laid out in detail, including quantum circuits. Moreover, the chapter also explains the state evolution of an isolated quantum system, as is predicted by the Schrödinger equation, as well as non-unitary irreversible operations such as measurement. More details of classical and quantum randomness and their mathematical representation is discussed, leading to the density matrix. representation of a quantum state.

This is the chapter that gets down to applying concepts from the previous chapters about qubits to construct a quantum computer. It teaches how numbers can be stored in quantum computers and how their functions can be evaluated. It also demonstrates the computational speed-up that quantum computers offer over their classical counterparts through the study of Deutsch, Deutsch-Jozsa, and Bernstein-Vazirani algorithms. Finally, it gives a practical demonstration of speed-up in search algorithms provided by Grover’s search algorithm.

Quantum entanglement requires a minimum of two quantum systems to exist, and each quantum system has to have a minimum of two levels. This is exactly what a two-qubit system is, which in this chapter is explored on various levels: state description, entanglement measures, useful theorems, quantum gates, hidden variable theory, quantum teleportation.

Deducing the quantum state of your device is essential for diagnosing and perfecting it, and the methods needed for this are introduced in this chapter. We also extend the discussion to methods used to validate noisy, intermediate-scale quantum computers when they grow too large for tomography to be used.

The generic properties of physical qubits are discussed in detail: in particular the need for an energy gap to ensure cooling and its implications for the size of devices. The basic notions of controlling qubits by external forces shows us how single-qubit gates are implemented.

Several other technologies under development to exploit quantum power are discussed in this chapter. You will learn about quantum key distribution; improving measurements of phase shifts is used as an example to demonstrate the power of entanglement in beating the standard quantum limit. How the latter is used to improve detection of objects is also discussed. Finally, modelling complicated quantum systems by designing simpler and easier to control systems, represented by quantum circuits, simplifies the studying of such systems, allowing us to gain better insight into their physics and to make better predictions about them.

Quantum computing technology was born in the 1970s and 1980s when a handful of visionary thinkers such as Paul Benioff, Richard Feynman, and David Deutsch first speculated about how the precepts of quantum mechanics might impact computer science. In 1984 Gilles Brassard, a computer scientist and cryptographer, and Charles Bennett, a specialist in physics and information theory, devised a practical application for quantum mechanics in the field of secure communication.

Here we build the skills needed to master how a quantum computer can factor very large numbers much more efficiently than a classical computer; i.e., it is a chapter dedicated to Shor’s algorithm. The Fourier transform, and its quantum analogue are introduced and applied to period finding. These are then applied to show how the problem of factoring large numbers amounts to finding the period of a modular exponential function. Moreover, the consequences of such a capability on the everyday security in (internet) communications using RSA encryption is also discussed.