This chapter covers quantum algorithmic primitives for loading classical data into a quantum algorithm. These primitives are important in many quantum algorithms, and they are especially essential for algorithms for big-data problems in the area of machine learning. We cover quantum random access memory (QRAM), an operation that allows a quantum algorithm to query a classical database in superposition. We carefully detail caveats and nuances that appear for realizing fast large-scale QRAM and what this means for algorithms that rely upon QRAM. We also cover primitives for preparing arbitrary quantum states given a list of the amplitudes stored in a classical database, and for performing a block-encoding of a matrix, given a list of its entries stored in a classical database.

This chapter covers the multiplicative weights update method, a quantum algorithmic primitive for certain continuous optimization problems. This method is a framework for classical algorithms, but it can be made quantum by incorporating the quantum algorithmic primitive of Gibbs sampling and amplitude amplification. The framework can be applied to solve linear programs and related convex problems, or generalized to handle matrix-valued weights and used to solve semidefinite programs.

This chapter covers quantum algorithmic primitives related to linear algebra. We discuss block-encodings, a versatile and abstract access model that features in many quantum algorithms. We explain how block-encodings can be manipulated, for example by taking products or linear combinations. We discuss the techniques of quantum signal processing, qubitization, and quantum singular value transformation, which unify many quantum algorithms into a common framework.

Some modern implementations of vector concepts rely heavily on a precise knowledge of time. Measurements of time, both ancient and modern, have always been heavily tied to Earth’s rotation, and so this rotation must be described in detail. I begin that task by describing Earth’s orientation relative to the solar system and the stars, and use a DCM to quantify Earth’s orientation at a given moment. This introduces the idea of Universal Time, UT1. Further concepts require a short discussion of relativity, both special and general, which I do by using a balloon to describe curved spacetime. The result is UTC, our modern ‘Greenwich Mean Time’. Measuring time over long periods is made easy through the concept of the Julian day, and so I discuss the Julian and Gregorian calendars. I include a detailed example of using these ideas to calculate the sight direction of a star at some time and place on Earth.

In the Preface, we motivate the book by discussing the history of quantum computing and the development of the field of quantum algorithms over the past several decades. We argue that the present moment calls for adopting an end-to-end lens in how we study quantum algorithms, and we discuss the contents of the book and how to use it.

The previous chapter described Earth’s orientation. I now build on that to construct orbital theory with a greater emphasis on vectors and coordinates than is traditional in that subject. I use Euler angles, rotation sequences, and the theory constructed around these in previous chapters to simplify what can often be a confusing barrage of notation in orbital theory. I include two very detailed examples here: sighting an Earth satellite and sighting Jupiter.

Rigid-body dynamics uses vectors heavily, and in particular the angular velocity vector described in a previous chapter. I derive the main quantities and results of the subject: angular momentum, moment of inertia, torque, and the relevant conservation laws. Examples are the spinning top and precessing bicycle wheel. I also provide a detailed calculation of Earth’s precession period arising from the gravity of the Sun and Moon.

Vehicle attitude is typically quantified by a DCM, a quaternion, or a triplet of Euler angles. I discuss how each of these objects changes with attitude by deriving the well-known time derivative of each. That requires the concept of angular velocity, which I discuss in detail. I end the chapter by describing why time derivatives of Euler angles cause so much confusion to many practitioners.

This chapter covers the quantum adiabatic algorithm, a quantum algorithmic primitive for preparing the ground state of a Hamiltonian. The quantum adiabatic algorithm is a prominent ingredient in quantum algorithms for end-to-end problems in combinatorial optimization and simulation of physical systems. For example, it can be used to prepare the electronic ground state of a molecule, which is used as an input to quantum phase estimation to estimate the ground state energy.

This chapter covers quantum linear system solvers, which are quantum algorithmic primitives for solving a linear system of equations. The linear system problem is encountered in many real-world situations, and quantum linear system solvers are a prominent ingredient in quantum algorithms in the areas of machine learning and continuous optimization. Quantum linear systems solvers do not themselves solve end-to-end problems because their output is a quantum state, which is one of its major caveats.

I start by invoking the ‘fundamental rule of calculus notation’ to ensure the correct translation of an English sentence into the language of calculus. As examples, I derive the ‘rocket equation’ and the standard expression for the gravitational potential of a sphere. I discuss the importance of treating units properly. I make the important point that a frame is not the same as a system of coordinates. I distinguish between ‘proper vectors’ and ‘coordinates vectors’, which is needed for a proper understanding of transforming coordinates. Because the study of vehicle attitude is built on basis vectors, I show how to construct these from both an intuitive viewpoint and a purely mathematical viewpoint.

This chapter presents an introduction to the theory of quantum fault tolerance and quantum error correction, which provide a collection of techniques to deal with imperfect operations and unavoidable noise afflicting the physical hardware, at the expense of moderately increased resource overheads.