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

7.5 The Bloch sphere

We describe the state of a qubit by a vector

a |0 + b |1 = r1 eϕ1 i |0 + r2 eϕ2 i |1
in C2 with r1 and r2 non-negative numbers in R.

The magnitudes r1 and r2 are related by r12 + r22 = 1. This is a mathematical condition. We saw in section 7.3 2 − ϕ1 that is significant and not the individual phases ϕ1 and ϕ2. This is a physical condition and it also means we can take a to be real.

We also saw that we could represent a quantum state as

|ψ = cos(θ/2) |0 + sin(θ/2) eϕ i |1.
We do this via a non-linear projection and a change of coordinates, and get a point on the surface of the Bloch sphere.

The two angles have the ranges 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π. θ is measured from the positive z axis and ϕ from the positive x axis in the xy...