This chapter covers the quantum algorithmic primitive called Gibbs sampling. Gibbs sampling accomplishes the task of preparing a digital representation of the thermal state, also known as the Gibbs state, of a quantum system in thermal equilibrium. Gibbs sampling is an important ingredient in quantum algorithms to simulate physical systems. We cover multiple approaches to Gibbs sampling, including algorithms that are analogues of classical Markov chain Monte Carlo algorithms.

I derive the important equation that relates the time derivative of a vector computed in one frame to that computed in another frame. I make the point that we must understand the distinction between frames and coordinates to appreciate what the equations are saying. That discussion leads naturally to the concept of centrifugal and Coriolis forces in rotating frames. I use the frame-dependent time derivative to derive some equations for robotics, and finish with a wider discussion of the time derivative for tensors and in fluid flow.

We are entering a new phase in the information revolution driven by the introduction of new artificial intelligence (AI) technologies and how they are being used to transform the data amassed within organisations over the previous 30 or so years. This revolution started hundreds of years ago when moveable type was used to print books at a scale and speed not previously possible. The rise of mass media and then mass communications in the form of telecommunication networks, coupled with the data processing capabilities of computers, powered the next phases. Thirty years ago, the expansion of the internet and the widespread adoption of the World Wide Web (WWW) as a means to publish and share information brought the digital revolution into our homes and businesses. Mobile computing has put powerful, always connected computers in the pockets of most people in the industrialised world and social networks have provided platforms for individuals to reach billions of others with their thoughts and ideas.

A result of these innovations and the ways they have been used is a world awash with information, most of which sits unused, unstructured and hidden away in archives and data silos within the organisations, public and private, that created it (Lange, 2023). Despite significant advances in information management techniques and technologies, only a fraction of the value of this data is being realised. However, we are now at an inflection point where much of the infrastructure is in place to begin the process of changing that.

This chapter covers applications of quantum computing in the area of nuclear and particle physics. We cover algorithms for simulating quantum field theories, where end-to-end problems include computing fundamental physical quantities and scattering cross sections. We also discuss simulations of nuclear physics, which encompasses individual nuclei as well as dense nucleonic matter such as neutron stars.

This chapter starts by showing that the DCM is a rotation matrix, and vice versa. I introduce Euler matrices as important examples of rotation matrices. I give examples extracting angle–axis information from a DCM. This chapter includes a study of what tensors are, and their role in this subject.

This chapter covers the quantum Fourier transform, which is an essential quantum algorithmic primitive that efficiently applies a discrete Fourier transform to the amplitudes of a quantum state. It features prominently in quantum phase estimation and Shor’s algorithm for factoring and computing discrete logarithms.

This chapter covers applications of quantum computing relevant to the financial services industry. We discuss quantum algorithms for the portfolio optimization problem, where one aims to choose a portfolio that maximizes expected return while minimizing risk. This problem can be formulated in several ways, and quantum solutions leverage methods for combinatorial or continuous optimization. We also discuss quantum algorithms for estimating the fair price of options and other derivatives, which are based on a quantum acceleration of Monte Carlo methods.

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.