Book Image

Dancing with Qubits

By : Robert S. Sutor
5 (1)
Book Image

Dancing with Qubits

5 (1)
By: Robert S. Sutor

Overview of this book

Quantum computing is making us change the way we think about computers. Quantum bits, a.k.a. qubits, can make it possible to solve problems that would otherwise be intractable with current computing technology. Dancing with Qubits is a quantum computing textbook that starts with an overview of why quantum computing is so different from classical computing and describes several industry use cases where it can have a major impact. From there it moves on to a fuller description of classical computing and the mathematical underpinnings necessary to understand such concepts as superposition, entanglement, and interference. Next up is circuits and algorithms, both basic and more sophisticated. It then nicely moves on to provide a survey of the physics and engineering ideas behind how quantum computing hardware is built. Finally, the book looks to the future and gives you guidance on understanding how further developments will affect you. Really understanding quantum computing requires a lot of math, and this book doesn't shy away from the necessary math concepts you'll need. Each topic is introduced and explained thoroughly, in clear English with helpful examples.
Table of Contents (16 chapters)
Preface
13
Afterword

5.9 Eigenvectors and eigenvalues

Let’s pause to review some of the wonderful features of diagonal matrices. Recall that a diagonal matrix has 0s everywhere except maybe on the main diagonal. A simple example for R3 is

display math
Its effect on the standard basis vectors e1, e2, and e3 is to stretch by a factor of 3 along the first, leave the second alone, and reflect across the xy-plane and then stretch by a factor of 2 along the third. A general diagonal matrix looks like
display math
Of course, we might be dealing with a small matrix and not have quite so many 0s. For a diagonal matrix D as above,
  • det(D) = d1 d2 ··· dn.
  • tr (D) = d1 + d2 + ··· + dn
  • DT = D.
  • D is invertible if and only if none of the di are 0.
  • If D is invertible,
    display math
  • If b1, ..., bn is the basis we are using, then D b1 = d1 b1, D b2 = d2 b2, ..., D bn = dn bn.

Focusing on this last...