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

10.5 Order and period finding

Consider the function ak on whole numbers k for a fixed a in N greater than 1. For example, if a = 3 then the first 12 values are

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As we look at larger exponents k, the values of 3k will just get larger and larger. If we instead use modular arithmetic as we saw in section 3.7 k cannot get arbitrarily large. For example, modulo M = 13, the values we get are
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Working modulo M = 16 yields
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Finally, for modulo M = 22 we get
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In each case, the sequence starts repeating. That is, the sequences, and hence the functions, are periodic. If we define fa(x) = ax mod M for a coprime to M, then the smallest positive integer r such that fa(x) = fa(x + r) for all x is called the period of fa.

Question 10.5.1

What happens when we work modulo M = 23, 24, and 25?

For M = 13 in the first example, the period r = 3. For M = 16, r = 4. In the final example...