This chapter delves into the application of trapped ions in electromagnetic fields for quantum computing, starting with the technique of confining ions using a linear Paul trap. It then examines the encoding of qubits within the ions’ electronic states. The interaction between an ion and a laser, pivotal for system operations, is analyzed next. This interaction underpins the initialization of ions via laser cooling and the execution of one- and two-qubit gates. The two-qubit gates also employ the ions’ motional states to extend beyond the traditional qubit space. The process also includes a method for measuring qubit states by detecting the photons released when ions are excited. The text identifies key sources of noise that can affect ion traps. It concludes with a summary and the advantages and challenges associated with trapped-ion quantum computing.

This chapter examines the use of photon ensembles for quantum computing. It opens with a primer on photons, normal modes, and both linear and nonlinear optics. The discussion then advances to the technologies employed in generating and detecting single photons, followed by methods of qubit encoding and initialization. Subsequently, the focus shifts to qubit control, detailing the execution of single-qubit gates using linear optical elements and the Knill–Laflamme–Milburn (KLM) protocol for two-qubit gates. While the textbook predominantly centers on the circuit model, alternative models of quantum computing – specifically, one-way quantum computing and continuous-variable quantum computing – and their optical implementations are introduced. Additionally, it outlines the primary sources of noise affecting these systems. The chapter wraps up with a reflection on the comparative benefits and limitations of optical quantum computing.

This chapter delves into superconducting qubits, starting with the essentials of superconductivity and circuit design. Central to this discussion is the Josephson junction, a key element in creating superconducting qubits. The text focuses on the transmon, the archetype in this field, while acknowledging other designs. Initialization of the transmon involves sophisticated dilution refrigerators, a process that is also examined. Additionally, the principles of circuit quantum electrodynamics (QED) are introduced as the framework for qubit control and measurement. Attention is then given to noise sources and their effect on superconducting qubits, with insights that apply to various qubit systems. The chapter wraps up by highlighting the strengths and challenges of superconducting qubits for quantum computing.

Chapter 2 serves as a primer on quantum mechanics tailored for quantum computing. It reviews essential concepts such as quantum states, operators, superposition, entanglement, and the probabilistic nature of quantum measurements. This chapter focuses on two-level quantum systems (i.e. qubits). Mathematical formulations that are specific to quantum mechanics are introduced, such as Dirac (bra–ket) notation, the Bloch sphere, density matrices, and Kraus operators. This provides the reader with the necessary tools to understand quantum algorithms and the behaviour of quantum systems. The chapter concludes with a review of the quantum harmonic oscillator, a model to describe quantum systems that are complementary to qubits and used in some quantum computer implementations.

This chapter explores the origin, key components, and essential concepts of quantum computing. It begins by charting the series of discoveries by various scientists that crystallized into the idea of quantum computing. The text then examines how certain applications have driven the evolution of quantum computing from a theoretical concept to an international endeavour. Additionally, the text clarifies the distinctions between quantum and classical computers, highlighting the DiVincenzo criteria, which are the five criteria for quantum computing. It also introduces the circuit model as the foundational paradigm for quantum computation. Lastly, the chapter sheds light on the reasons for the belief that quantum computers are more powerful than classical ones (touching on quantum computational complexity) and physically realizable (touching on quantum error correction).

The third chapter examines the capabilities of liquid-state NMR systems for quantum computing. It begins by grounding the reader in the basics of spin dynamics and NMR spectroscopy, followed by a discussion on the encoding of qubits into the spin states of the nucleus of atoms inside molecules. The narrative progresses to describe the implementation of single-qubit gates via external magnetic fields, weaving in key concepts such as the rotating-wave approximation, the Rabi cycle, and pulse shaping. The technique for orchestrating two-qubit gates, leveraging the intrinsic couplings between the spins of nuclei of atoms within a molecule, is subsequently detailed. Additionally, the chapter explains the process of detecting qubits’ states through the collective nuclear magnetization of the NMR sample and outlines the steps for qubit initialization. Attention then shifts to the types of noise that affect NMR quantum computers, shedding light on decoherence and the critical T1 and T2 times. The chapter wraps up by providing a synopsis, evaluating the strengths and weaknesses of liquid-state NMR for quantum applications, and a note on the role of entanglement in quantum computing.

The final chapter details some methods for evaluating the performance of quantum computers. It begins by delineating the essential features of quantum benchmarks and organizes them into a three-tiered framework. Initially, it discusses early-stage benchmarks that provide a detailed analysis of basic operations on a few qubits, emphasizing fidelity tests and tomography. Then, it progresses to intermediate-stage benchmarks that provide a more generalized appraisal of gate quality, circuit depth, and length. Concluding the benchmarking spectrum, later-stage benchmarks are introduced, aimed at evaluating the overall reliability and efficiency of quantum computers operating with a large number of qubits (e.g. 1000 or more).