This chapter examines quantum decoherence, a process by which quantum information is lost due to environmental interactions. Various noise channels, such as bit-flip, phase-flip, and depolarizing channels, are discussed to illustrate common errors in qubit states. The Kraus representation and Lindblad equation offer frameworks for modeling these interactions. Metrics such as T1 (relaxation time) and T2 (decoherence time) are introduced to measure qubit stability. Understanding decoherence mechanisms is critical for developing strategies to preserve quantum information, laying the groundwork for quantum error correction techniques and highlighting the challenges in creating reliable quantum systems.

This chapter covers quantum error correction, essential for preserving quantum information in the presence of noise. It introduces the bit-flip and phase-flip codes as foundational error-correction methods, building toward Shor’s code, which corrects general single-qubit errors. Logical qubits are formed by encoding physical qubits to maintain stability. Stabilizer codes are presented as a systematic framework for error correction, enabling fault-tolerant quantum computing. These principles are crucial for creating scalable quantum systems that can perform reliable computations, even in noisy environments, addressing a central challenge in quantum computing’s practical implementation.

This chapter explores classical computation fundamentals, starting with Turing machines as a foundation for defining computability. The universal Turing machine is introduced, emphasizing the theoretical basis for all computable functions. Computational complexity is discussed, differentiating between tractable and intractable problems and explaining complexity classes as a framework for problem-solving. The chapter also covers the circuit model, providing a bridge between theoretical constructs and modern computer architecture. Finally, the concept of reversible computation is introduced, which has implications for energy-efficient processing. Through these topics, the chapter delineates classical computation’s limitations, setting up the motivation to transition into quantum approaches in subsequent chapters.

This book is about approximately solving problems of the form.

The algorithm based on gradient descent in the previous chapter is simple and computationally efficient, at least provided the projection can be computed. There are two limitations, however.

The purpose of this chapter is to introduce the necessary tools from optimisation, convex geometry and convex analysis. You can safely skip this chapter, referring back as needed. The main concepts introduced are as follows:

We already saw an application of exponential weights to linear and quadratic bandits in Chapter 7. The same abstract algorithm can also be used for convex bandits but the situation is more complicated. Throughout this chapter we assume the losses are bounded and there is no noise:

Function classes like Fb are non-parametric. In this chapter we shift gears by studying two important parametric classes: Fb,lin and Fb,quad. The main purpose of this chapter is to use the machinery designed for linear bandits to prove an upper bound on the minimax regret for quadratic bandits. On the positive side the approach is both elementary and instructive. More negatively, the resulting algorithm is not computationally efficient. Before the algorithms and regret analysis we need three tools: covering numbers, optimal experimental design and the exponential weights algorithm.

This chapter introduces quantum computation by comparing classical and quantum computers. Core concepts including qubits, superposition, and entanglement are introduced, setting the stage for deeper exploration. Various quantum computing models are summarized, with a focus on the circuit and topological models. The chapter explains why quantum computing is necessary, especially for tasks beyond classical computing’s limits. It discusses existing quantum platforms and provides an overview of their capabilities and limitations. The chapter also offers a brief historical perspective, touches on computational energy efficiency, and forecasts a quantum future where quantum and classical computing work in tandem. This groundwork provides essential insights into quantum computation’s potential and upcoming chapters’ explorations of algorithmic and theoretical principles.

Like the bisection method (Chapter 4), cutting plane methods are most naturally suited to the stochastic setting. For the remainder of the chapter we assume the setting is stochastic and the loss function is bounded: