Matrix theory play a fundamental role in classical and quantum computing. Among this paper, we study the fundamental concepts of some matrices including unitary matrix and Hermitian matrix, which may convert some initial qubits into various kinds of qubits. We also explore the fundamental concept of eigenvalues. Furthermore, we input the concept of product vectors. Finally we use above-mentioned concepts examing the entanglement witness for verifying the entanglement in quantum computing and circuits.
Matrix theory play a fundamental role in classical and quantum computing. Among this paper, we study the fundamental concepts of some matrices including unitary matrix and Hermitian matrix, which may convert some initial qubits into various kinds of qubits. We also explore the fundamental concept of eigenvalues. Furthermore, we input the concept of product vectors. Finally we use above-mentioned concepts examing the entanglement witness for verifying the entanglement in quantum computing and circuits.
Quantum mechanics has proven its position as the primary theory for accurately describing microscopic domains. Numerous fields in physics utilize quantum mechanics to great effects, with one prominent application being quantum computing. Quantum computing extends the field of quantum mechanics to information theory, striving to perform computational tasks utilizing quantum phenomena with notably greater efficiency compared to classical alternatives. In this thesis, I will go through the basics required in the field of quantum computing. To understand quantum computing demands adapting a new type of matrix mechanical view. This involves considering Hilbert spaces, which are linear vector space having a well defined inner product. Hilbert spaces contain state vectors describing quantum states, as well as operators for altering and measuring these states. The dynamics of quantum systems are governed by three quantum postulates. These specify Hilbert spaces and state vectors as the fundamental components of quantum systems, Hermitian operators as providing the physical manifestation of these systems, and finally, the Schrödinger's equation to describe the time evolution of such systems. These notions are required in producing quantum computing. Quantum computing uses qubits as the computational basis, analogous to classical bits but with the added capability of quantum superposition. Qubits are represented as two-state vectors, and can be physically realized with any quantum-behaving binary-valued system, like the electron spins. Qubits can be combined into higher dimensional states with the use of tensor products, causing the system's Hilbert space to expand as 2^n with n being the number of qubits, allowing for exponential computational speedup. Computational logic can be constructed by acting on the qubits using unitary operators, known as quantum gates. The combinations of qubits and quantum gates can be visualized as quantum circuits, depicted using specialized circuit diagram notations. An essential part of quantum computing, and the phenomena enabling exponential computational speedup, is quantum entanglement. Entanglement arises in interacting qubits, intertwining them so that the individual states become correlated. Entanglement can be characterized by being unable to reduce a combined state into its pure constituents. The amount of entanglement in a system can be measured using von Neumann entropy, analogous to Shannon entropy in classical information theory. Entanglement serves various purposes in quantum computing and is a defining feature distinguishing it from the classical domain. In this thesis, the entanglement properties of a periodic four-qubit circuit with alternating quantum gates were simulated. The simulation was performed computationally, and measured entanglement entropy as a function of the amount of circuit periods. The results demonstrated different characteristics of entanglement depending on the parameters used for the quantum gates, with the entanglement either being periodic, non-occurring or chaotic in nature.
Quantum computing uses a mathematical language that can seem unfamiliar to readers from programming, engineering and other technical backgrounds. Qubits are represented by vectors, quantum gates by matrices, measurement outcomes by eigenvalues, and probability amplitudes by complex numbers. Quantum Computing Preliminaries develops this language carefully, connecting each mathematical idea with its computational and physical meaning. The book begins with complex vector spaces and explains why quantum states require more than classical probabilities. It introduces qubits, superposition, normalisation, basis vectors and Dirac bra–ket notation, supported by worked calculations and geometric interpretation using the Bloch sphere. The discussion then turns to inner products and the geometry of quantum states. Readers learn how to calculate norms, test orthogonality, determine state overlap and apply the Born rule. Particular attention is given to complex conjugation and the distinction between the ordinary dot product and the Hermitian inner product used in quantum mechanics.The treatment of matrix operators establishes the roles of linear, unitary and Hermitian operators. It explains matrix multiplication, non-commutativity, eigenvalues, eigenvectors, reversibility and norm preservation. The Pauli and Hadamard gates provide concrete examples that link matrix algebra to rotations, the Bloch sphere, quantum-state evolution, and measurement. Outer products and projection operators are developed as tools for constructing operators, selecting state components and describing ideal projective measurements. The book examines idempotence, completeness, repeated measurement and the distinction between the ket–bra outer product and the tensor product used to form composite quantum systems. The final chapter explores changes of basis, operator representations and diagonalisation. It shows how the same physical state or operation can have different numerical representations in the computational and Hadamard bases. Readers derive the similarity transformation, use outer-product substitution, diagonalise Hermitian operators and interpret the spectral decomposition geometrically. Examples involving the Pauli-X, Pauli-Y, Pauli-Z and Hadamard operators demonstrate how basis choice exposes phase, population and measurement structure without changing the underlying physics. Throughout the book, Python and NumPy examples translate the mathematics into executable calculations. The programs demonstrate complex-vector operations, inner products, eigendecomposition, basis transformation and numerical verification while explaining the importance of using operations suited to complex Hilbert spaces. Each chapter includes conceptual, mathematical, computational and challenge exercises. Detailed solutions provide step-by-step derivations, program output and physical interpretation, allowing the book to support both guided study and independent practice. Quantum Computing Preliminaries is intended for students, software developers, engineers and technically minded readers preparing to study quantum circuits, algorithms, simulation, measurement, entanglement or error correction. It assumes familiarity with elementary algebra and programming but does not require prior training in quantum mechanics. Its purpose is to provide a clear mathematical foundation for understanding how quantum information is represented, transformed and measured.
The solution of sets of linear simultaneous equations can be accomplished using matrices and we show how this is done. In quantum computation, matrices ‘operate’ or act on vectors to move a system from one quantum state to another as it evolves with time. The solution of sets of linear simultaneous equations can be accomplished using matrices and we show how this is done. Matrices can also be used to perform transformations of points in two- and three-dimensional space through rotations and reflections. Self-adjoint matrices play an important role in quantum computation because, as we shall see, they have real eigenvalues which correspond to the measured values of quantum observables. The procedure for calculating the inverse of a general n×n matrix necessitates the introduction of some new terminology. The rows of a unitary matrix are pairwise orthogonal so that the inner product of two distinct rows will be zero.
This chapter briefly describes the basic concepts and principles of quantum computing. Firstly, the concepts of qubit, quantum coherence, quantum decoherence, quantum entanglement, quantum density operators, linear operators, inner products, outer products, tensor products, Hermite operators, and unitary operators are described. Then, the four basic assumptions of quantum mechanics are introduced, focusing on the measurement assumptions of quantum mechanics. Finally, the definition of commonly used quantum logic gates is given including single qubit gates, double qubit gates, and multiple qubit gates. These contents provide the necessary theoretical basis for subsequent chapters.
This chapter represents our first step into quantum information theory (QIT). The key to operating such a transition is to become familiar with the concept of the quantum bit, or qubit , which is a probabilistic superposition of the classical 0 and 1 bits. In the quantum world, the classical 0 and 1 bits become the pure states |0〉 and |1〉, respectively. It is as if a coin can be classically in either heads or tails states, but is now allowed to exist in a superposition of both! Then I show that qubits can be physically transformed by the action of unitary matrices , which are also called operators . I show that such qubit transformations, resulting from any qubit manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere . The Pauli matrices are shown to generate all such unitary transformations. These transformations are reversible , because they are characterized by unitary matrices ; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable “zoo” of unitary operators, called I , X , Y , Z , H , CNOT, CCNOT, CROSSOVER or SWAP, controlled- U , and controlled-controlled- U . These gates can be used to form quantum circuits , involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product , as progressively introduced through the above description, is eventually formalized.
Quantum computing plays a major role in modern computation. It can perform operations exponentially faster when compared to classical computation. It has applications in various areas like secure communication, drug design, artificial intelligence, cyber security, etc. Thus, the researchers and students are showing interest to perform experiments in quantum computing to design novel architectures. But to learn/understand quantum computing, one should have strong knowledge of its basics. Because quantum computing performs operations at the atomic level, so the learners need to understand basic concepts like Qubits, superposition, quantum gates, etc. This chapter gives a clear idea about the basic concepts of quantum computing like Qubits, superposition, entanglement, and quantum gates.