Trigonometry is the basis of the book’s subject. I begin with length and angle, and then generalise to coordinates. This requires the important idea of a directed angle, which enables us to relate the sine and cosine of an angle to coordinates in any given orientation of a set of axes. I discuss the details of inverting the sine/cosine/tangent functions, and introduce a new function name to replace the inappropriate name “atan2” that often appears in the literature. The chapter ends with examples of calculating bearing and elevation.

Results of previous chapters come together here in the equations that model a vehicle’s position and attitude given a knowledge of, for example, its angular turn rates. These equations can seem perplexing at first glance, and so I derive them in careful steps, again making strong use of vectors and the frame dependence of the time derivative. I end with a detailed example of applying these equations to a spinning top.

This chapter covers the quantum algorithmic primitive of Hamiltonian simulation, which aims to digitally simulate the evolution of a quantum state forward in time according to a Hamiltonian. There are several approaches to Hamiltonian simulation, which are best suited to different situations. We cover approaches for time-independent Hamiltonian simulation based on product formulas, the randomized compiling approach called qDRIFT, and quantum signal processing. We also discuss a method that leverages linear combination of unitaries and truncation of Taylor and Dyson series, which is well suited for time-dependent Hamiltonian simulation

This chapter provides an overview of how to perform a universal set of logical gates on qubits encoded with the surface code, via a procedure called lattice surgery. This is the most well-studied approach for practical fault-tolerant quantum computation. We perform a back-of-the-envelope end-to-end resource estimation for the number of physical qubits and total runtime required to run a quantum algorithm in this paradigm. This provides a method for converting logical resource estimates for quantum algorithms into physical resource estimates.

This chapter covers the quantum algorithmic primitive called quantum phase estimation. Quantum phase estimation is an essential quantum algorithmic primitive that computes an estimate for the eigenvalue of a unitary operator, given as input an eigenstate of the operator. It features prominently in many end-to-end quantum algorithms, for example, computing ground state energies of physical systems in the areas of condensed matter physics and quantum chemistry. We carefully discuss nuances of quantum phase estimation that appear when it is applied to a superposition of eigenstates with different eigenvalues.

This chapter covers applications of quantum computing in the area of continuous optimization, including both convex and nonconvex optimization. We discuss quantum algorithms for computing Nash equilibria for zero-sum games and for solving linear, second-order, and semidefinite programs. These algorithms are based on quantum implementations of the multiplicative weights update method or interior point methods. We also discuss general quantum algorithms for convex optimization which can provide a speedup in cases where the objective function is much easier to evaluate than the gradient of the objective function. Finally, we cover quantum algorithms for escaping saddle points and finding local minima in nonconvex optimization problems.

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.