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Essential Mathematics for Quantum Computing

Essential Mathematics for Quantum Computing

By : Leonard S. Woody III
4.6 (19)
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Essential Mathematics for Quantum Computing

Essential Mathematics for Quantum Computing

4.6 (19)
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)
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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

Subspaces

Let's say you have a set, U, of vectors and it is a subset of a set, V, of vectors (U V). This situation is shown in the following diagram:

Figure 4.1 – The set U as a subset of V

Is it possible that U is a subspace of V? Well, yes. It has met the first condition for subspaces, namely, that the potential subspace has to be a subset of the bigger vector space's set of vectors. What's next? Well, U also has to be a vector space using the same field as the vector space V. This seems like it might be an exhaustive thing to do, but it has been proven that we only need to check for three small conditions to make sure U is a subspace of V, and two of them have to do with the closure property from Chapter 3, Foundations. As a reminder, here it is:

  • Closure: For every a,b A, a ֎ b produces an element, c, that is also in the set A. f: A × A → A

Armed with the concept of a subset and closure...

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