I introduce an important way to think about and construct a DCM: by implementing a yaw–pitch–roll sequence of rotations on a model aircraft. This does away with the widespread but rather involved method of describing the relative orientation of two axis sets by drawing them with a common origin. For this, we must distinguish the idea of a rotation in a sequence being about either a ‘space-fixed’ axis or a ‘carried-along’ axis. Users of these terms tend to fall into two groups, ‘active’ and ‘passive’. I state the ‘fundamental theorem of rotation sequences’, which does away with any need for the reader to stand in one group or the other. I also discuss the extraction of Euler angles from a DCM, and examine infinitesimal rotations. I discuss two methods of interpolating from an initial to a final orientation; one of these is used widely in computer graphics, but both methods must be discussed for the computer-graphics method to be understood. I end with a calculation of the position and attitude of a robot arm.

This chapter covers the quantum algorithmic primitives of amplitude amplification and amplitude estimation. Amplitude amplification is a generalization of Grover’s quantum algorithm for the unstructured search problem. Amplitude estimation can be understood in a similar framework, where it utilizes quantum phase estimation to estimate the value of the amplitude or probability associated with a quantum state. Both amplitude amplification and amplitude estimation provide a quadratic speedup over their classical counterparts, and feature prominently as an ingredient in many end-to-end algorithms.

This chapter covers applications of quantum computing in the area of quantum chemistry, where the goal is to predict the physical properties and behaviors of atoms, molecules, and materials. We discuss algorithms for simulating electrons in molecules and materials, including both static properties such as ground state energies and dynamic properties. We also discuss algorithms for simulating static and dynamic aspects of vibrations in molecules and materials.

This chapter covers applications of quantum computing in the area of condensed matter physics. We discuss algorithms for simulating the Fermi-Hubbard model, which is used to study high-temperature superconductivity and other physical phenomena. We also discuss algorithms for simulating spin models such as the Ising model and Heisenberg model. Finally, we cover algorithms for simulating the Sachdev-Ye-Kitaev (SYK) model of strongly interacting fermions, which is used to model quantum chaos and has connections to black holes.

This chapter covers applications of quantum computing in the area of combinatorial optimization. This area is related to operations research, and it encompasses many tasks that appear in science and industry, such as scheduling, routing, and supply chain management. We cover specific problems where a quadratic quantum speedup may be available via Grover’s quantum algorithm for unstructured search. We also cover several more recent proposals for achieving superquadratic speedups, including the quantum adiabatic algorithm, the quantum approximate optimization algorithm (QAOA), and the short-path algorithm.

An important set of coordinates to understand is that of our oblate Earth. I derive the equations transforming latitude/longitude/height to and from the ECEF cartesian axes. I use the model aircraft of a previous chapter as an aid to visualise the rotation sequences that are useful for calculating NED or ENU coordinates at a given point on or near Earth’s surface. I use these in a detailed example of sighting a distant aircraft. This leads to a description of the ‘DIS standard’ designed for such scenarios. I also use these ideas in a detailed example of estimating Earth’s gravity at a given point, which is necessary for implementing inertial navigation systems.

The initial excitement as well as considerable hype from software companies, AI developers and technology commentators following the launch of ChatGPT and other GenAI products in late 2022 and into 2023 has died down to a large extent. The ‘magic’ of seeing text and images being created in seconds from a few simple prompts is now just another clever thing that computers can do and is becoming part of many people's daily workflows. It was the same with e-mail, the World Wide Web (WWW), mobile phones and social media when they first became available. They are all now part of the warp and weft of everyday life. In time, GenAI and the applications that incorporate it will be no different. However, ‘time’ is the watchword here. This will not happen overnight for all the reasons discussed in this book. Developers need to demonstrate the value AI offers to organisations through real use cases and solid evidence of a return on investment. Adopting organisations need to be confident that the benefits outweigh the risks and this requires further work from developers and vendors in removing problems such as hallucinations and privacy breaches. If agentic AI is to take hold, then trust in such systems will be key. Alongside this, regulators and public policy makers will need to adapt their approaches as the technology evolves and its opportunities and risks become clearer. Finally, education will be a vital factor in helping workers, existing and yet to enter the workforce, adapt to this transformative technology, as well as teaching all individuals what they can and cannot trust online. This last requirement is, perhaps, the most important as it touches on foundational issues such as literacy and democracy.

A short chapter that describes the book’s content. It covers the core principles, and discusses some ways in which the book’s description of them differs from that of less technical descriptions.

This chapter covers variational quantum algorithms, which act as a primitive ingredient for larger quantum algorithms in several application areas, including quantum chemistry, combinatorial optimization, and machine learning. Variational quantum algorithms are parameterized quantum circuits where the parameters are trained to optimize a certain cost function. They are often shallow circuits, which potentially makes them suitable for near-term devices that are not error corrected.

I introduce quaternions by recounting the story of how Hamilton discovered them, but in far more detail than other authors give. This detail is necessary for the reader to understand why Hamilton wrote his quaternion equations in the way that he did. I describe the role of quaternions in rotation, show how to convert between them and matrices, and discuss their role in modern computer graphics. I describe a modern problem in detail whereby Hamilton’s original definition has been ‘hijacked’ in a way that has now produced much confusion. I end by describing how quaternions play a role in topology and quantum mechanics.

This chapter covers a number of disparate applications of quantum computing in the area of machine learning. We only consider situations where the dataset is classical (rather than quantum). We cover quantum algorithms for big-data problems relying upon high-dimensional linear algebra, such as Gaussian process regression and support vector machines. We discuss the prospect of achieving a quantum speedup with these algorithms, which face certain input/output caveats and must compete against quantum-inspired classical algorithms. We also cover heuristic quantum algorithms for energy-based models, which are generative machine learning models that learn to produce outputs similar to those in a training dataset. Next, we cover a quantum algorithm for the tensor principal component analysis problem, where a quartic speedup may be available, as well as quantum algorithms for topological data analysis, which aim to compute topologically invariant properties of a dataset. We conclude by covering quantum neural networks and quantum kernel methods, where the machine learning model itself is quantum in nature.