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

Linear combinations

Once we have established that we can add our vectors and multiply them by scalars, we can start to talk about linear combinations. Linear combinations are just the scaling and addition of vectors to form new vectors. Let's start with our two vectors we have been working with the whole time, |a and |b. I want to scale my vector |a by two to get a new vector |c, as shown in the following screenshot:

Figure 1.6 – |a⟩ scaled by two to produce |c⟩

Figure 1.6 – |a scaled by two to produce |c

As we have said, we can do this algebraically as well, as the following equation shows:

Then, I want to take my vector |b and scale it by two to get a new vector, |d, as shown in the following screenshot:

Figure 1.7 – |b⟩ scaled by two to produce |d⟩

Figure 1.7 – |b scaled by two to produce |d

So, now, we have a vector |c that is two times |a, and a vector |d that is two times |b:

Can I add these two new vectors, |c and |d? Certainly! I will do that, but I will express |e as a linear combination of |a and |b in the following way:

Vector |e is a linear combination of vectors |a and |b! Now, I can show this all geometrically, as follows:

Figure 1.8 – Linear combination

Figure 1.8 – Linear combination

This can also be represented in the following equation:

So, we now have a firm grasp on Euclidean vectors, the algebra you can perform with them, and the concept of a linear combination. We will use that in this next section to describe a quantum phenomenon called superposition.