Essential Mathematics for Quantum Computing

Essential Mathematics for Quantum Computing

by Leonard S. Woody III
Released April 2022
Publisher(s): Packt Publishing
ISBN: 9781801073141

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Book description

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!

Table of contents

  1. Essential Mathematics for Quantum Computing
  2. Acknowledgements
  3. Contributors
  4. About the author
  5. About the reviewers
  6. Preface
    1. How to use this book
    2. Who this book is for
    3. What this book is not
    4. Download the color images
    5. Conventions used
    6. Get in touch
    7. Share Your Thoughts
  7. Section 1: Introduction
  8. Chapter 1: Superposition with Euclid
    1. Vectors
      1. Vector addition
      2. Scalar multiplication
    2. Linear combinations
    3. Superposition
      1. Measurement
    4. Summary
    5. Answers to exercises
      1. Exercise 1
      2. Exercise 2
  9. Chapter 2: The Matrix
    1. Defining a matrix
      1. Notation
      2. Redefining vectors
    2. Simple matrix operations
      1. Addition
      2. Scalar multiplication
      3. Transposing a matrix
    3. Defining matrix multiplication
      1. Multiplying vectors
      2. Matrix-vector multiplication
      3. Matrix multiplication
      4. Properties of matrix multiplication
    4. Special types of matrices
      1. Square matrices
      2. Identity matrices
    5. Quantum gates
      1. Logic gates
      2. Circuit model
    6. Summary
    7. Answers to exercises
      1. Exercise 1
      2. Exercise 2
      3. Exercise 3
      4. Exercise 4
      5. Exercise 5
    8. References
  10. Section 2: Elementary Linear Algebra
  11. Chapter 3: Foundations
    1. Sets
      1. The definition of a set
      2. Notation
      3. Important sets of numbers
      4. Tuples
      5. The Cartesian product
    2. Functions
      1. The definition of a function
      2. Exercise 1
      3. Invertible functions
    3. Binary operations
      1. The definition of a binary operation
      2. Properties
    4. Groups
    5. Fields
      1. Exercise 2
    6. Vector space
    7. Summary
    8. Answers to Exercises
      1. Exercise 1
      2. Exercise 2
    9. Works cited
  12. Chapter 4: Vector Spaces
    1. Subspaces
      1. Definition
      2. Examples
      3. Exercise 1
    2. Linear independence
      1. Linear combination
      2. Linear dependence
    3. Span
    4. Basis
    5. Dimension
    6. Summary
    7. Answers to exercises
      1. Exercise 1
  13. Chapter 5: Using Matrices to Transform Space
    1. Linearity
    2. What is a linear transformation?
      1. Describing linear transformations
    3. Representing linear transformations with matrices
      1. Matrices depend on the bases chosen
      2. Matrix multiplication and multiple transformations
      3. The commutator
    4. Transformations inspired by Euclid
      1. Translation
      2. Rotation
      3. Projection
      4. Exercise two
    5. Linear operators
    6. Linear functionals
    7. A change of basis
    8. Summary
    9. Answers to exercises
      1. Exercise one
      2. Exercise two
    10. Works cited
  14. Section 3: Adding Complexity
  15. Chapter 6: Complex Numbers
    1. Three forms, one number
      1. Definition of complex numbers
    2. Cartesian form
      1. Addition
      2. Multiplication
      3. Exercise 1
      4. Complex conjugate
      5. Absolute value or modulus
      6. Division
      7. Powers of i
    3. Polar form
      1. Polar coordinates
      2. Exercise 3
      3. Defining complex numbers in polar form
      4. Example
      5. Multiplication and division in polar form
      6. Example
      7. De Moivre's theorem
    4. The most beautiful equation in mathematics
    5. Exponential form
      1. Exercise 4
      2. Conjugation
      3. Multiplication
      4. Example
    6. Conjugate transpose of a matrix
    7. Bloch sphere
    8. Summary
    9. Exercises
      1. Exercise 1
      2. Exercise 2
      3. Exercise 3
      4. Exercise 4
    10. References
  16. Chapter 7: EigenStuff
    1. The inverse of a matrix
    2. Determinants
      1. Exercise one
    3. The invertible matrix theorem
    4. Calculating the inverse of a matrix
      1. Exercise two
    5. Eigenvalues and eigenvectors
      1. Definition
      2. Example with a matrix
      3. The characteristic equation
      4. Finding eigenvectors
      5. Multiplicity
    6. Trace
    7. The special properties of eigenvalues
    8. Summary
    9. Answers to exercises
      1. Exercise one
      2. Exercise two
  17. Chapter 8: Our Space in the Universe
    1. The inner product
    2. Orthonormality
      1. The norm
      2. Orthogonality
      3. Orthonormal vectors
      4. The Kronecker delta function
    3. The outer product
      1. Exercise two
    4. Operators
      1. Representing an operator using the outer product
      2. Exercise 3
      3. The completeness relation
      4. The adjoint of an operator
    5. Types of operators
      1. Normal operators
      2. Hermitian operators
      3. Unitary operators
      4. Projection operators
      5. Positive operators
    6. Tensor products
      1. The tensor product of vectors
      2. Exercise four
      3. The basis of tensor product space
      4. Exercise five
      5. The tensor product of operators
      6. Exercise six
      7. The inner product of composite vectors
      8. Exercise seven
    7. Summary
    8. Answers to exercises
      1. Exercise one
      2. Exercise two
      3. Exercise three
      4. Exercise four
      5. Exercise five
      6. Exercise six
      7. Exercise seven
  18. Chapter 9: Advanced Concepts
    1. Gram-Schmidt
    2. Cauchy-Schwarz and triangle inequalities
    3. Spectral decomposition
      1. Diagonal matrices
      2. Spectral theory
      3. Example
      4. Bra-ket notation
      5. Example take two
    4. Singular value decomposition
    5. Polar decomposition
    6. Operator functions and the matrix exponential
    7. Summary
    8. Works cited
  19. Section 4: Appendices
  20. Appendix 1: Bra–ket Notation
    1. Operators
      1. Bras
  21. Appendix 2: Sigma Notation
    1. Sigma
      1. Variations
    2. Summation rules
  22. Appendix 3: Trigonometry
    1. Measuring angles
      1. Degrees
      2. Radians
    2. Trigonometric functions
    3. Formulas
    4. Summary
    5. The trig cheat sheet
      1. Pythagorean identities
      2. Double angle identities
      3. Sum/difference identities
      4. Product-to-sum identities
    6. Works cited
  23. Appendix 4: Probability
    1. Definitions
    2. Random variables
      1. Discrete random variables
      2. The measures of a random variable
    3. Summary
    4. Works cited
  24. Appendix 5: References
    1. Why subscribe?
  25. Other Books You May Enjoy
    1. Packt is searching for authors like you
    2. Share Your Thoughts

Product information

  • Title: Essential Mathematics for Quantum Computing
  • Author(s): Leonard S. Woody III
  • Release date: April 2022
  • Publisher(s): Packt Publishing
  • ISBN: 9781801073141