Essential Mathematics for Quantum Computing: A beginner’s guide to just the math you need without needless complexities
- Length: 252 pages
- Edition: 1
- Language: English
- Publisher: Packt Publishing
- Publication Date: 2022-04-22
- ISBN-10: 1801073147
- ISBN-13: 9781801073141
- Sales Rank: #68692 (See Top 100 Books)
Demystify quantum computing by learning the math it is built on
Key Features
- Build a solid mathematical foundation to get started with developing powerful quantum solutions
- Understand linear algebra, calculus, matrices, complex numbers, vector spaces, and other concepts essential for quantum computing
- Learn the math needed to understand how quantum algorithms function
Book Description
Quantum computing is an exciting subject that offers hope to solve the world’s most complex problems at a quicker pace. It is being used quite widely in different spheres of technology, including cybersecurity, finance, and many more, but its concepts, such as superposition, are often misunderstood because engineers may not know the math to understand them. This book will teach the requisite math concepts in an intuitive way and connect them to principles in quantum computing.
Starting with the most basic of concepts, 2D vectors that are just line segments in space, you’ll move on to tackle matrix multiplication using an instinctive method. Linearity is the major theme throughout the book and since quantum mechanics is a linear theory, you’ll see how they go hand in hand. As you advance, you’ll understand intrinsically what a vector is and how to transform vectors with matrices and operators. You’ll also see how complex numbers make their voices heard and understand the probability behind it all.
It’s all here, in writing you can understand. This is not a stuffy math book with definitions, axioms, theorems, and so on. This book meets you where you’re at and guides you to where you need to be for quantum computing. Already know some of this stuff? No problem! The book is componentized, so you can learn just the parts you want. And with tons of exercises and their answers, you’ll get all the practice you need.
What you will learn
- Operate on vectors (qubits) with matrices (gates)
- Define linear combinations and linear independence
- Understand vector spaces and their basis sets
- Rotate, reflect, and project vectors with matrices
- Realize the connection between complex numbers and the Bloch sphere
- Determine whether a matrix is invertible and find its eigenvalues
- Probabilistically determine the measurement of a qubit
- Tie it all together with bra-ket notation
Who this book is for
If you want to learn quantum computing but are unsure of the math involved, this book is for you. If you’ve taken high school math, you’ll easily understand the topics covered. And even if you haven’t, the book will give you a refresher on topics such as trigonometry, matrices, and vectors. This book will help you gain the confidence to fully understand quantum computation without losing you in the process!
Essential Mathematics for Quantum Computing Acknowledgements Contributors About the author About the reviewers Preface How to use this book Who this book is for What this book is not Download the color images Conventions used Get in touch Share Your Thoughts Section 1: Introduction Chapter 1: Superposition with Euclid Vectors Vector addition Scalar multiplication Linear combinations Superposition Measurement Summary Answers to exercises Exercise 1 Exercise 2 Chapter 2: The Matrix Defining a matrix Notation Redefining vectors Simple matrix operations Addition Scalar multiplication Transposing a matrix Defining matrix multiplication Multiplying vectors Matrix-vector multiplication Matrix multiplication Properties of matrix multiplication Special types of matrices Square matrices Identity matrices Quantum gates Logic gates Circuit model Summary Answers to exercises Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise 5 References Section 2: Elementary Linear Algebra Chapter 3: Foundations Sets The definition of a set Notation Important sets of numbers Tuples The Cartesian product Functions The definition of a function Exercise 1 Invertible functions Binary operations The definition of a binary operation Properties Groups Fields Exercise 2 Vector space Summary Answers to Exercises Exercise 1 Exercise 2 Works cited Chapter 4: Vector Spaces Subspaces Definition Examples Exercise 1 Linear independence Linear combination Linear dependence Span Basis Dimension Summary Answers to exercises Exercise 1 Chapter 5: Using Matrices to Transform Space Linearity What is a linear transformation? Describing linear transformations Representing linear transformations with matrices Matrices depend on the bases chosen Matrix multiplication and multiple transformations The commutator Transformations inspired by Euclid Translation Rotation Projection Exercise two Linear operators Linear functionals A change of basis Summary Answers to exercises Exercise one Exercise two Works cited Section 3: Adding Complexity Chapter 6: Complex Numbers Three forms, one number Definition of complex numbers Cartesian form Addition Multiplication Exercise 1 Complex conjugate Absolute value or modulus Division Powers of i Polar form Polar coordinates Exercise 3 Defining complex numbers in polar form Example Multiplication and division in polar form Example De Moivre's theorem The most beautiful equation in mathematics Exponential form Exercise 4 Conjugation Multiplication Example Conjugate transpose of a matrix Bloch sphere Summary Exercises Exercise 1 Exercise 2 Exercise 3 Exercise 4 References Chapter 7: EigenStuff The inverse of a matrix Determinants Exercise one The invertible matrix theorem Calculating the inverse of a matrix Exercise two Eigenvalues and eigenvectors Definition Example with a matrix The characteristic equation Finding eigenvectors Multiplicity Trace The special properties of eigenvalues Summary Answers to exercises Exercise one Exercise two Chapter 8: Our Space in the Universe The inner product Orthonormality The norm Orthogonality Orthonormal vectors The Kronecker delta function The outer product Exercise two Operators Representing an operator using the outer product Exercise 3 The completeness relation The adjoint of an operator Types of operators Normal operators Hermitian operators Unitary operators Projection operators Positive operators Tensor products The tensor product of vectors Exercise four The basis of tensor product space Exercise five The tensor product of operators Exercise six The inner product of composite vectors Exercise seven Summary Answers to exercises Exercise one Exercise two Exercise three Exercise four Exercise five Exercise six Exercise seven Chapter 9: Advanced Concepts Gram-Schmidt Cauchy-Schwarz and triangle inequalities Spectral decomposition Diagonal matrices Spectral theory Example Bra-ket notation Example take two Singular value decomposition Polar decomposition Operator functions and the matrix exponential Summary Works cited Section 4: Appendices Appendix 1: Bra–ket Notation Operators Bras Appendix 2: Sigma Notation Sigma Variations Summation rules Appendix 3: Trigonometry Measuring angles Degrees Radians Trigonometric functions Formulas Summary The trig cheat sheet Pythagorean identities Double angle identities Sum/difference identities Product-to-sum identities Works cited Appendix 4: Probability Definitions Random variables Discrete random variables The measures of a random variable Summary Works cited Appendix 5: References Why subscribe? Other Books You May Enjoy Packt is searching for authors like you Share Your Thoughts
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