Book Image

Essential Mathematics for Quantum Computing

By : Leonard S. Woody III
5 (1)
Book Image

Essential Mathematics for Quantum Computing

5 (1)
By: Leonard S. Woody III

Overview of this book

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.
Table of Contents (20 chapters)
1
Section 1: Introduction
4
Section 2: Elementary Linear Algebra
8
Section 3: Adding Complexity
13
Section 4: Appendices
Appendix 1: Bra–ket Notation
Appendix 2: Sigma Notation
Appendix 5: References

Exponential form

Complex numbers written in terms of e are said to be in the exponential form, as opposed to the polar or Cartesian form we have seen earlier. Using Euler's formula, we can express a complex number, z, as:

So

As you can see, the exponential form is very close to polar form, but now you have θ in one place instead of two!

Exercise 4

Express the following complex numbers in exponential form:

Conjugation

As we have seen, the conjugation of a complex number is represented as a reflection around the real axis. For complex numbers in exponential form, this means we just change the sign of the angle to get the complex conjugate:

Multiplication

Multiplication and division are even easier in exponential form and are one of the reasons why it is so preferred to work with. We can take our steps for multiplication from the polar form and easily restate them in exponential form.

Given the two complex numbers...