The purpose of this chapter is to introduce and analyse the surrogate loss functions used in Chapters 10 and 11. The results are stated in as much generality as possible to facilitate their use in future applications. In case you want a quick summary of the results, read this introductory section for the basic definitions and then head directly to Section 12.9.

This chapter presents key quantum mechanics principles essential for understanding quantum computation. The postulates of quantum mechanics, mixed states, and density matrices are introduced, along with the Stern–Gerlach experiment’s role in illustrating quantum behavior. Topics such as quantum coherence, entanglement, and the EPR paradox are covered to clarify the fundamental distinctions between classical and quantum systems. Measurement is explored with an emphasis on positive operator-valued measures (POVM), a key concept in understanding quantum state collapse. These principles provide a foundation for studying quantum computation and are essential for understanding qubit behavior, quantum information processing, and subsequent algorithmic structures.

This chapter delves into the quantum circuit model, a primary framework for quantum computation. It begins with the qubit, exploring its representation on the Bloch sphere and its probabilistic measurement outcomes. Quantum gates are introduced as the basic operational units, transforming qubits via unitary operations. The chapter discusses single- and two-qubit gates, building up to universal quantum computation, which enables any quantum function to be constructed through a finite set of gates. This chapter provides an in-depth understanding of information processing in quantum circuits, establishing a practical foundation for executing quantum algorithms and advancing to topics like entanglement-based operations and fault-tolerant design in later chapters.

This chapter introduces seminal quantum algorithms that illustrate quantum computation’s efficiency over classical methods. The Deutsch and Deutsch–Jozsa algorithms showcase quantum parallelism, offering solutions to specific problems with fewer computational steps. The quantum Fourier transform (QFT) is introduced, underpinning period-finding algorithms as well as Shor’s algorithm for integer factorization, which has major implications for cryptography. Grover’s algorithm demonstrates a quadratic speedup for unstructured search problems. By using superposition, entanglement, and phase manipulation, these algorithms highlight the computational power of quantum mechanics and its potential to outperform classical techniques, particularly for complex or classically intractable tasks.

In this chapterwe introduce an idea that is ubiquitous in zeroth-order optimisation, which is to use a gradient-based algorithm but replace the true gradients with estimated gradients of a smoothed loss. Except for Section 5.4, we assume throughout this chapter that the constraint set contains a euclidean ball of unit radius, the losses are bounded, Lipschitz and there is no noise:

This chapter briefly outlines the key algorithmic ideas and history of bandit convex optimisation. There follow in Section 2.4 and Section 2.5 summary tables of known lower and upper bounds for the various settings studied in this book.

Submodular functions are sometimes viewed as a combinatorial analogue of convexity via a gadget called the Lovászwi extension that we explain in a moment.

This chapter introduces topological quantum computation (TQC), a model using non-Abelian anyons, specifically Fibonacci anyons, for information processing via braiding operations. The braid group and fusion rules are central to TQC, enabling operations that remain robust against certain environmental errors. TQC provides inherent fault tolerance, reducing susceptibility to local disturbances. The chapter concludes by examining the challenges and future potential of topological models, marking TQC as a promising, albeit complex, path toward scalable and robust quantum computing solutions.

This chapter delves into topological order, a phase of matter with implications for quantum computation. The ℤ2 toric code model is introduced, using lattice arrangements of qubits to demonstrate topological protection against errors. Anyons, particles exhibiting unique exchange statistics, are utilized for encoding information through braiding operations. Surface codes are discussed as practical implementations of topological error correction, leveraging topological entanglement entropy to protect quantum information. This approach provides a highly resilient framework for quantum error correction, essential for developing fault-tolerant quantum computers with intrinsic stability against certain types of errors.

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.