Essential Mathematics for Quantum Computing

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Cover Copyright Contributors Table of Contents Preface 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 Index Other Books You May Enjoy