What this book covers

The book is split into two parts reflecting the natural progression from analog to digital quantum computing, with an increasing depth in the analysis and understanding of algorithms. However, we start with a chapter that covers the basic principles of quantum mechanics and provides the motivation for the computational methods based on those principles.

Chapter 1, The Principles of Quantum Mechanics, covers the basic mathematical principles of quantum mechanics. It provides the necessary definitions and discusses the postulates of quantum mechanics and their relevance to quantum computing.

Part I: Analog Quantum Computing – Quantum Annealing

For a number of years, quantum annealers were the only large-scale quantum computing devices available for experiments in solving non-trivial NP-hard combinatorial optimisation problems. Although quantum annealing specifically targets solving classically hard optimisation problems, it can also be used for many different hybrid quantum-classical problems, such as samplers and classifiers. The book provides detailed coverage of these applications and illustrates them on specific financial use cases.

Chapter 2, Adiabatic Quantum Computing, introduces the concept of analog quantum computing. The chapter starts with the principles of adiabatic quantum computing and proceeds with the quantum adiabatic theorem. The physical realisation of adiabatic quantum computing is quantum annealing, which is explained alongside its classical counterpart – simulated annealing. The chapter also discusses the implementation, limitations, and universality of adiabatic quantum computing.

Chapter 3, Quadratic Unconstrained Binary Optimisation, describes the single most important application of quantum annealing: solving classically hard optimisation problems. A wide range of combinatorial optimisation problems can be formulated as Quadratic Unconstrained Binary Optimisation (QUBO) problems (or, equivalently, as Ising problems) solvable on a quantum annealer. The chapter provides in-depth coverage of the forward and reverse quantum annealing techniques and demonstrates the power of quantum annealing on a discrete portfolio optimisation use case.

Chapter 4, Quantum Boosting, extends the range of QUBO applications beyond combinatorial optimisation and outlines the quantum boosting algorithm designed to combine a large number of weak classical classifiers into a strong classifier. The algorithm is formulated as a QUBO problem executable on a quantum annealer and applied to the use case of building a strong predictor of credit card defaults from a large number of weak predictors.

Chapter 5, Quantum Boltzmann Machine, explores further machine learning applications of quantum annealing. The quantum Boltzmann machine can be used as a generative model for sampling from a learned probability distribution as well as an efficient method of pre-training deep feedforward neural networks.

Part II: Gate Model Quantum Computing

Gate model quantum computing hardware has seen enormous progress in recent years and is quickly approaching the quantum advantage threshold. The search for quantum advantage – the real-world productive application of a quantum computing solution that outperforms any viable classical alternative – is one of the strongest motivations for quantum computing research in finance and elsewhere. The book explores the main quantum computing algorithms implementable on existing NISQ devices and highlights a range of possible financial applications that may benefit from this new computing paradigm.

Chapter 6, Qubits and Quantum Logic Gates, introduces the paradigm of gate model quantum computing. We start with the basic concepts of classical digital computing and expand the computational logic to accommodate the new principles of superposition and entanglement. The chapter draws parallels between and contrasts classical and quantum logic gates and shows how to assemble quantum circuits from individual quantum logic gates.

Chapter 7, Parameterised Quantum Circuits and Data Encoding, proceeds with the construction of quantum algorithms covering both the theoretical and the practical aspects of building Parameterised Quantum Circuits (PQCs) and demonstrates how classical samples can be encoded into quantum states processed by the PQCs. The chapter provides a detailed description of specific data encoding techniques.

Chapter 8, Quantum Neural Network, considers parameterised quantum circuits trained as classifiers. Throughout this chapter, we show how differentiable and non-differentiable learning algorithms can be used to efficiently train quantum neural networks. The chapter also discusses the limitations of existing QPUs and how to design quantum circuits that extract maximum benefit from the available quantum computing hardware. We investigate QNN performance on a credit approval use case and benchmark it against several standard classical classifiers.

Chapter 9, Quantum Circuit Born Machine, introduces a quantum counterpart to classical generative models such as Boltzmann machines – the Quantum Circuit Born Machine (QCBM). The chapter starts with the definition of the QCBM and how it can be efficiently configured and run on available QPUs, continues with the differentiable and non-differentiable learning and training procedures, and concludes with the market generator use case benchmarked against the classical Restricted Boltzmann Machine.

Chapter 10, Variational Quantum Eigensolver, introduces the variational principle and formulates the Variational Quantum Eigensolver (VQE) approach to optimisation problems. The chapter discusses a hybrid quantum-classical approach to training the VQE and looks at the practical aspects of running it on NISQ devices.

Chapter 11, Quantum Approximate Optimisation Algorithm, describes the gate model quantum computing approach (inspired by quantum annealing) to solving QUBO-type problems, such as NP-hard Max-Cut optimisation problems.

Chapter 12, Quantum Kernels and Quantum Two-Sample Test, covers two cutting-edge QML algorithms based on quantum feature maps realised with the help of parameterised quantum circuits.

Chapter 13, The Power of Parameterised Quantum Circuits, investigates the main sources of quantum advantage we expect to demonstrate on practical applications of parameterised quantum circuits. The chapter focuses on two elements: strong regularisation provided by quantum neural networks and the expressive power of quantum generative models.

Chapter 14, Advanced QML Models, discusses new promising quantum algorithms and techniques such as quantum GAN, Bayesian quantum circuit, symmetric encryption, and quantum semi-definite programming.

Chapter 15, Beyond NISQ, looks beyond the capabilities of NISQ era computers. The chapter presents several algorithms that lie in the foundation of many envisaged future applications of quantum computers. The algorithms include quantum Fourier transform, quantum phase estimation, quantum Monte Carlo, and quantum linear solver.