The origin of decoherence of qubits is described by a simple example, and the two key methods to defeat decoherence, namely decoherence-free spaces and error-correcting codes are introduced.

Here we discuss some of the interesting paradigm shifts that have been proposed for quantum computers: namely, using pseudo-pure states, cluster states, and non-deterministic gates.

After discussing the divorce of configuration and observable that is characteristic of the quantum description of reality, the reader is introduced to the awesome potential computational power that is afforded by quantum computation.

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.