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.

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.

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 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.

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.

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.

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.

A discussion of realization spaces, including an example of an oriented matroid with disconnected extension space, is provided. In the later part, a proof of the Universality Theorem and a discussion of some of its consequences follows.

The chapters addresses the various axiomatizations and the equivalences between them and presents an introduction to the Plucker relations. The chapter finishes with some discussion of nonrealizable oriented matroids and the impossibility of an excluded minor characterization.

The geometric motivation for the theory is combinatorial data associated with matrices, vector arrangements, hyperplane arrangements, and subspaces of real vector spaces. Interpretations of this data are given in terms of linear algebra, discrete geometry, and the Plucker embedding of the Grassmannian. Elementary proofs of cryptomorphisms for realizable oriented matroids are provided. The chapter finishes with an application of Gale Diagrams.

The chapters provides a survey on the topology of various posets of oriented matroids analogous to various topological spaces, including extension spaces, combinatorial Grassmannians, and combinatorial flag spaces. A general framework for interpreting maps from spaces to posets is laid down, by way of McCord’s Theorem and the Semi-algebraic Triangulation Theorem. The chapter includes a discussion of the (now-disproved) extension space conjecture and of the various results on the topology of the MacPhersonian.