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.

The origin of decoherence of qubits is described by a simple example, and the two key methods to defeat decoherence, namely decoherence-free spaces and error-correcting codes are introduced.

Here we discuss some of the interesting paradigm shifts that have been proposed for quantum computers: namely, using pseudo-pure states, cluster states, and non-deterministic gates.

After discussing the divorce of configuration and observable that is characteristic of the quantum description of reality, the reader is introduced to the awesome potential computational power that is afforded by quantum computation.

The universe we live in is both strange and interesting. This strangeness comes about because, at the most fundamental level, the universe is governed by the laws of quantum mechanics. This is the most spectacularly accurate and powerful theory ever devised, one that has given us insights into many aspects of the world, from the structure of matter to the meaning of information. This textbook provides a comprehensive account of all things quantum. It starts by introducing the wavefunction and its interpretation as an ephemeral wave of complex probability, before delving into the mathematical formalism of quantum mechanics and exploring its diverse applications, from atomic physics and scattering, to quantum computing. Designed to be accessible, this volume is suitable for both students and researchers, beginning with the basics before progressing to more advanced topics.

The Hamiltonian plays the starring role in the standard formulation of quantum mechanics. But, back in the classical world, there are two equivalent ways to write down a theory, one using the Hamiltonian and the other using the Lagrangian. It’s natural to wonder if there might also be another formulation of quantum mechanics, where things are written in terms of the Lagrangian. Happily, there is. And it’s lovely. Its called the path integral

Over the past hundred years or so, physicists have developed a foolproof and powerful tool that allows us to understand everything and anything in the universe. You take the object that you’re interested in and you throw something at it. Ideally, you throw something at it really hard. This technique was developed around the turn of the 20th century and has since allowed us to understand everything from the structure of atoms, to the structure of materials, to the structure of DNA. In short, throwing stuff at other stuff is the single most important experi- mental method available to science. Because of this, it is given a respectable sounding name. We call it scattering.

This chapter explores what we could do with a computer whose operating system is quantum mechanics, rather than classical mechanics. One of the answers is: factorise primes really quickly. We will explain why this is interesting.

Theres a lot of interesting physics to be found if you subject an atom to an electric or magnetic field. This chapter explores this physics. It covers the Stark effect and the Zeeman effect and Rabi oscillations. it also looks at what happens when coherent states of photons in a cavity interact with atoms.