Abstract This chapter introduces the four postulates of quantum mechanics: state space, evolution, and composition – which are associated with isolated quantum mechanical systems, and measurement, which defines what happens when information is extracted from a quantum mechanical system. Several important theoretical results concerning measurement are given, including the violation of the Bell inequalities, which establishes the profoundly non-classical nature of entanglement. This chapter also introduces qubits, which constitute the building blocks for the majority of quantum algorithms, and the Bloch sphere is presented as a convenient way of picturing a single-qubit state.
Abstract This chapter introduces the four postulates of quantum mechanics: state space, evolution, and composition – which are associated with isolated quantum mechanical systems, and measurement, which defines what happens when information is extracted from a quantum mechanical system. Several important theoretical results concerning measurement are given, including the violation of the Bell inequalities, which establishes the profoundly non-classical nature of entanglement. This chapter also introduces qubits, which constitute the building blocks for the majority of quantum algorithms, and the Bloch sphere is presented as a convenient way of picturing a single-qubit state.
This presentation provides a comprehensive overview of the foundational principles of quantum computation. It begins by contrasting classical and quantum information, highlighting how quantum bits (qubits) leverage superposition, entanglement, and interference to process information in ways that surpass classical limits. The document explains the four postulates of quantum mechanics: state space (qubits as unit vectors in a Hilbert space), time evolution (via unitary operators), measurement (probabilistic collapse of the state), and composite systems (tensor products of subsystems). Key concepts such as the Bloch sphere representation, quantum gates (e.g., Hadamard, CNOT, Pauli gates), and quantum circuits are discussed, illustrating how quantum operations manipulate qubit states. The presentation also covers quantum evolution, the gate model for computation, and the universality of certain gate sets, emphasizing the potential for quantum parallelism and the transformative impact of quantum computing on fields like cryptography and optimization.
Abstract This chapter reviews all the rules of quantum mechanics and their mathematical notation. It also considers quantum entanglement, a fundamental resource in quantum information. There are four basic postulates of quantum mechanics, that indicate how to represent physical systems, how to represent observations, how to carry out measurements, and how systems evolve when “not measured”. Before describing the laws of quantum mechanics, their mathematical background is first described, along with a Mach Zehnder interferometer experiment in which the (strange) readings on the detectors can be explained by quantum mechanics. This chapter also discusses the Dirac notation, qubits, Hilbert spaces, projective measurements and operations, unitary operations, eigenvectors and eigenvalues, spectral decomposition, applications of the spectral theorem, Dirac notation shorthands, and mixed states.
In general, all forms of information processing can be considered in a quantum mechanical context, including for example communication, channel capacities, and the quantum limits to information extraction. In this chapter we introduce the basic features of quantum information processing. We first present qubits as abstractions of two-level addressable quantum systems, before generalizing this concept to higher-dimensional qudit systems, and continuous variables. We introduce the stabilizer formalism towards the beginning of our discussion as a convenient way to track quantum information. In Section 2.2 we derive the no-cloning theorem, which leads us to cryptography, teleportation, and quantum repeaters. Section 2.3 describes quantum computing with qubits. We start with a brief description of the circuit model, and use it to define cluster states and the one-way model. We study the properties of cluster states and discuss error correction. In the last section we introduce quantum computing over continuous variables. Quantum information Classically, information is carried by ‘bits’, which are physical systems that have two (macroscopic) states denoted by 0 and 1. These bits are described classically, which immediately raises the question of how we should extend this information-theoretic description to the quantum mechanical case. This will lead to the quantum unit of information, the quantum bit or ‘qubit’. 2.1.1 Classical and quantum bits We can define the qubit as a two-dimensional quantum system, which means that the state space of the system is a two-dimensional complex Hilbert space ℋ.
Quantum computing represents a fundamentally new paradigm in information processing, in which information is encoded and manipulated using two-level quantum systems known as qubits, whose evolution is governed by the principles of quantum mechanics. This chapter presents a structured introduction to the theoretical foundations of quantum computation, beginning with the limitations of classical computation and the physical constraints that motivate the need for a quantum model of computation. We develop the mathematical framework required to describe quantum information, introducing qubits as state vectors in Hilbert space and examining the roles of superposition, measurement, and unitary evolution. Single-qubit operations are presented through the Pauli and Hadamard gates, along with their geometric interpretation on the Bloch sphere. Building on this formalism, the discussion is extended to multi-qubit systems using tensor products, and quantum entanglement is introduced as a non-classical correlation fundamental to quantum information processing. The chapter then explores quantum algorithms, focusing on Grover’s unstructured search algorithm, to demonstrate how superposition, interference, and entanglement can be harnessed to achieve computational advantages over classical approaches. Finally, the chapter discusses applications of quantum computation, including quantum simulation, cryptography, search and optimization problems, and near-term hybrid quantum algorithms such as the Variational Quantum Eigensolver (VQE)and the Quantum Approximate Optimization (QAO) Algorithm, highlighting their relevance in the noisy intermediate-scale quantum (NISQ) era.
In this section we provide a broad overview of the basic principles of quantum mechanics and how they enable modern quantum technologies. The starting point is the concept of superposition, which allows quantum systems to be in multiple states at the same time and forms the basis of quantum computation using qubits and parallel processing of information. The conversation then turns to quantum entanglement, a non-classical effect that creates strong correlations between particles, and has applications in secure communication and quantum teleportation. Also mentioned is the quantum coherence, which preserves phase relations between quantum states and makes it possible for quantum systems to function correctly. But it is also emphasized that quantum coherence is fragile to environmental perturbations. This chapter also covers quantum measurement, taking into account the probabilistic nature of quantum mechanics, and how the act of observation results in the collapse of a quantum state to a definite outcome. These principles together constitute the theoretical basis of quantum computing, where quantum gates and algorithms and qubits take advantage of superposition, entanglement and interference to offer computational benefits over classical systems. The most important algorithms like Shor’s and Grover’s algorithms that demonstrate a considerable speed-up in solving the problem are discussed along with different models of quantum computing, such as gate, adiabatic and measurement-based models. Furthermore, the section reviews some physical realizations of quantum systems, such as superconducting qubits, trapped ions, photonic systems and topological qubits, with their advantages and difficulties. Overall, this discussion shows how the basic principles of quantum mechanics are the foundation for the development of advanced quantum technologies, while also dealing with the practical difficulties that arise in their implementation.