This chapter covers quantum interior point methods, which are quantum algorithmic primitives for application to convex optimization problems, particularly linear, second-order, and semidefinite programs. Interior point methods are a successful classical iterative technique that solve a linear system of equations at each iteration. Quantum interior point methods replace this step with quantum a quantum linear system solver combined with quantum tomography, potentially offering a polynomial speedup.

Books on vehicle attitude and motion often use tensors in their analyses, and I have discussed the reasons for that in a previous chapter. But tensors also carry an esotericism arising from being used to quantify the curved spacetime of general relativity. And so I end the book by telling the inquisitive reader how tensors ‘work’ more generally, and how this more advanced topic makes quick work of calculating the gradient, divergence, laplacian, and curl of vector calculus. I end with a discussion of parallel transport, which has found its way into the exotic ‘wander azimuth’ axes used in some navigation systems.

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.

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.