Matrices are widely, and extensively, used in quantum mechanics. Indeed, one of the avenues to quantum mechanics, that of Born, Heisenberg, and Jordan, is via matrices. One additional asset of matrices is that they are the natural mathematical language to describe quantum computing. Furthermore, matrices identify themselves in the physical tangible world via optical elements such as mirrors, beam splitters, polarization rotators, and interferometers. And these are the elements that comprise the hardware of photonic quantum computers. Here, we introduce the Pauli matrices https://www.w3.org/1998/Math/MathML" display="inline"> σ x , σ y , σ z , plus the identity matrix I , via the probability amplitudes of quantum entanglement. Once this is accomplished, we show how these matrices can be used to perform mathematical transformations, on quantum entanglement states, that form the bases of quantum computing as treated in the next chapter.
Matrices are widely, and extensively, used in quantum mechanics. Indeed, one of the avenues to quantum mechanics, that of Born, Heisenberg, and Jordan, is via matrices. One additional asset of matrices is that they are the natural mathematical language to describe quantum computing. Furthermore, matrices identify themselves in the physical tangible world via optical elements such as mirrors, beam splitters, polarization rotators, and interferometers. And these are the elements that comprise the hardware of photonic quantum computers. Here, we introduce the Pauli matrices https://www.w3.org/1998/Math/MathML" display="inline"> σ x , σ y , σ z , plus the identity matrix I , via the probability amplitudes of quantum entanglement. Once this is accomplished, we show how these matrices can be used to perform mathematical transformations, on quantum entanglement states, that form the bases of quantum computing as treated in the next chapter.
All computers are quantum computers but some are more quantum than others. Universal classical computers rely on the transistor, a quantum semiconductor, for their operation but use simple binary digital logic based on bits, 1 and 0. Quantum computers, those we are interested in here, utilize the qbits https://www.w3.org/1998/Math/MathML" display="inline"> | 1 〉 and https://www.w3.org/1998/Math/MathML" display="inline"> | 0 〉 , which are quantum states, to realize their quantum logic. These qbits are used in conjunction with superposition probability amplitudes, such as https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 + , https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 − , https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 + , and https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 − , to interact with optical components such as beam splitters, interferometers, and polarization rotators to perform quantum logical operations. In this chapter, a basic description of quantum logic, and quantum optical configurations, utilized in workable quantum computers is given.
Following an introduction on interferometric physical computers, the discussion turns to Feynman’s contribution to quantum computing and qbits. The ψ + , ψ − , ψ + , and ψ − quantum entanglement states are expressed in terms of the σ x , σ y , σ z , and I matrices and the discussion continues, on quantitative terms, Pauli gates, the Hadamard gate, and the CNOT gate. The equivalence between σ x and the polarization rotational matrix R is noted. Interaction between the ψ + , ψ − , ψ + , and ψ − states, R and the Hadamard gate give rise to whole families of quantum mathematical operations. Physical representations for R and the Hadamard gate are explicitly identified.
Using the vector direct product , the probability amplitudes for quantum entanglement, ψ + , ψ − , ψ + , and ψ − are expressed in a matrix form and shown to be represented by the Pauli matrices plus the identity matrix. The Hadamard matrix is derived from a beam splitter Hong-Ou-Mandel (HOM) configuration. Numerous mathematical operations with the polarization rotator matrix ( R = σ x ) and the Hadamard matrix operating on quantum entanglement states are demonstrated. This chapter concludes with a discussion of reversibility in quantum mechanics between quantum interference and quantum entanglement.
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.
Quantum teleportation, because of the word t eleportation , may seem to bring us to the edge of science fiction. But not so fast. Quantum teleportation refers to the use of a quantum entanglement apparatus with its entangled photon pair source, its two detection stages, plus additional necessary optics, to transport a quantum state from one site to another site. This is done via the use of quantum entanglement probability amplitudes and with the help of a classical transmission of a unitary transformation . A unitary transformation is a mathematical matrix that takes the form of https://www.w3.org/1998/Math/MathML" display="inline"> I , σ x , σ z , or i σ y , where I is the identity matrix and the Greek characters refer to the Pauli matrices.