Preface
Quantum Machine Learning – Most overhyped and underestimated field at the same time
Iordanis Kerenidis
Introduction
Why Quantum Computing? Quantum computing and AI will revolutionise and disrupt our society in the same way as classical digital computing did in the second half of the 20th century and the internet did in the first two decades of the 21st century.
Quantum computing (or, more generally, Quantum Information Theory) has been the subject of extensive research since the 1960s, but it was only in the last decade that progress on the hardware side has made it possible to test quantum computing algorithms; and it was only in the last several years that quantum computing’s supremacy was finally claimed as an experimental fact (i.e., a landmark experiment conducted on Google’s 53-qubit Sycamore quantum chip [16]).
The story of quantum computing is, in this respect, similar to the story of AI: AI was born in the 1950s but then experienced two “winters”, when interest in AI and Machine Learning declined considerably (following the Lighthill report in the UK and the Speech Understanding Research debacle in the US in the 1970s, and the LISP collapse in the 1990s), before becoming widely used and adopted to the point that we can no longer imagine our life without it.
Even though we cannot rule out a “quantum computing winter” before quantum computing technology becomes embedded in everyday life to the same extent as the internet, smartphones, and AI, the whole range of quantum computing breakthroughs we have witnessed in the last few years makes it somewhat unlikely.
With recent advances in the field, we have finally reached the era of Noisy Intermediate-Scale Quantum (NISQ) computing [237]. NISQ-era computers are powerful enough to test quantum computing algorithms and solve non-trivial real-world problems – and establish quantum speedup and quantum advantage over comparable classical hardware.
However, it is likely that the first real-world production-level business applications will be a hybrid quantum-classical protocol, where most of the computation and data processing is done classically, but the hardest problems are outsourced to the quantum chip. In finance, discrete portfolio optimisation problems, which are NP-hard, are such examples and clear objectives to tackle.
Why Quantum Machine Learning? It is a combination of quantum computing and AI that will likely generate the most exciting opportunities, including a whole range of possible applications in finance, but also in medicine, chemistry, physics, etc. We have already witnessed the first promising results achieved with Parameterised Quantum Circuits trained as either generative models (such as Quantum Circuit Born Machine, which can be used as a synthetic data generator) or discriminative models (such as a Quantum Neural Network that can be trained as a classifier). The possible use cases include market generators, data anonymisers, credit scoring, and the generation of trading signals.
All the models and techniques mentioned so far rely on the existence of universal, gate model quantum computers. However, there is another type of quantum hardware – quantum annealers – which realise the principle of adiabatic quantum computing. Quantum annealers are analog quantum computers that are very well suited for solving complex optimisation problems that are NP-hard for classical computers. Optimisation problems form a large class of hard-to-solve financial problems, not to mention the fact that many supervised and reinforcement learning tools used in finance are trained via solving optimisation problems (minimisation of a cost function, maximisation of reward).
An example of discriminative machine learning problems solved using quantum annealers includes building a strong classifier from several weak ones – the Quantum Boosting algorithm. The strong classifier is highly resilient against overtraining and against errors in the correlations of the physical observables in the training data. The quantum annealing-trained classifiers perform comparably to the state-of-the-art classical machine learning methods. However, in contrast to these methods, the annealing-based classifiers are simple functions of directly interpretable experimental parameters with clear physical meaning and demonstrate some advantage over traditional machine learning methods for small training datasets.
Another application of quantum annealing is in generative learning. In Deep Learning, a well-known approach for training a deep neural network starts with training a generative Deep Boltzmann Machine, typically using the Contrastive Divergence (CD) algorithm, then fine-tuning the weights using backpropagation or other discriminative techniques. However, generative training is often time consuming due to the slow mixing of Boltzmann (Gibbs) sampling. The quantum sampling-based training approach can achieve comparable or better accuracy with significantly fewer iterations of generative training than conventional CD-based training.
The main focus of this book is therefore on tackling practical real-world applications of Quantum Machine Learning (QML) algorithms executable on NISQ hardware rather than adopting the more traditional quantum computing textbook approach, diligently describing standard quantum computing algorithms (Shor’s, Grover’s, …), the quantum hardware demands of which are well beyond the capabilities of NISQ computers. The focus is also on the hybrid quantum-classical computational protocols that reflect the most productive way of harnessing the power of quantum computing – it is in tandem with classical computing that quantum computing solutions can provide maximum benefits to the users.
In this book, we cover all major QML algorithms that have been the subject of intensive research by the industry and that have shown early signs of potential quantum advantage. We also provide a balanced view of both analog and digital quantum computers and do not try to make a call on which quantum computing technology (superconducting qubits, trapped ions, neutral atoms, etc.) will be the eventual winner. The material is presented in a hardware-agnostic way with a strong emphasis on the fundamental characteristics of the algorithms rather than their hardware realisations, although we do not ignore the question of algorithms’ embedding and the practical limitations of the existing quantum computing hardware.
Why Finance? It is reasonable to expect that the incredibly fast rate of quantum hardware improvements we have witnessed over the last several years will lead to the widespread adoption of quantum computing techniques in finance. The finance industry is already investigating the potential of QML to solve classically hard practical problems and assist in achieving digital transformation. We might have moved past the point of quantum computing supremacy, but our quest to establish quantum computing advantage has just begun.
Quantitative finance is a discipline rich in interesting but computationally hard problems. Many such problems are interdisciplinary in nature and often require the transformation and adoption of mathematical and computational techniques developed in other fields. Here, we can mention, for example, the application of the theory of stochastic differential equations to option pricing [226], methods of optimal control theory to management science and economics [260], machine learning techniques to portfolio construction, and optimisation [193].
This is why we turn to finance when we are looking for a wide range of real-world use cases to test (and improve!) quantum computing algorithms. The book provides many examples of the quantum computing techniques and algorithms applied to solving practical financial problems such as portfolio optimisation, credit card default prediction, credit approvals, and generation of synthetic market data. At the same time, the methods and techniques are formulated and presented in the most general form – we hope our readers will discover many new exciting quantum computing use cases in finance and beyond.
Who this book is for
The book is primarily aimed at three main groups: academic researchers and STEM students; finance professionals working in the field of quantitative finance and related areas; computer scientists and ML/AI experts. At the same time, the book is organised in such a way as to be accessible and useful to a much wider audience.
The book does not require any prior knowledge of quantum mechanics and the complexity of the mathematical apparatus should not feel intimidating: although we do not sacrifice mathematical rigour, the emphasis is very much on the understanding of the fundamental properties of the models and algorithms.
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 demonstrated 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 quantum 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 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 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, 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 13, Looking Ahead, discusses new promising quantum algorithms and techniques such as the quantum kernel method, quantum GAN, Bayesian quantum circuit, and quantum semi-definite programming.
To get the most out of this book
This book is intended as an in-depth introduction to the power of quantum computing techniques for quantitative finance problems. While it is designed as self-contained, this book assumes that the reader has some familiarity with basic mathematical concepts in algebra, analysis, and computing. Knowledge of quantum mechanics is not required, and the main tools thereof shall be explained and made accessible to non-physicists.
Conventions used
There are a number of text conventions used throughout this book.
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When we wish to draw your attention to a particular definition or notation, the relevant lines or items are set in either italic or bold.
Important remarks and conclusions are shown in boxes.
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