How To Code For Quantum Computers

How to code for Quantum Computers

First Edition

By: Nívio dos Santos

Copyright © 2024 Nívio dos Santos

ISBN nº 978-65-01-17239-2

My dearest Fátima, your unwavering support and belief in me have been the driving force behind this book. Thank you for encouraging me to pursue my dreams and for always being there to celebrate my victories, no matter how small. Your love and encouragement have made all the difference.

Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let's try to generalize it so that the numbers we used to call probabilities can be negative numbers. As such, the theory could have been invented by mathematicians in the 19th century without any input from experiment. It wasn't, but it could have been.

by Scott Aaronson

Table of Contents

Chapter 1

Introduction to Quantum Computing

Required background

Code examples

Cirq

Conventional Bit vs Qubit

Double-slit experiment

Bra Ket Notation

Probabilities

What Are Complex Numbers?

Why are amplitudes complex Numbers?

Quick view in a Qubit

What more?

Real-Valued vs. Complex-Valued Quantum Theories

Quantum Entanglement

Entanglement Exchange Experiment

Navascués' Proposal

Conclusion about Complex Numbers

Unitary matrix

Chapter 2

Visual representation

Bloch Sphere

Where does this equation come from?

Circle Notation

Circle Notation for multi-Qubit scenarios

Choosing the Right Tool

Chapter 3

Qubits and Quantum Gates

Basic Gates with a single Qubit

The Identity Gate

Pauli Gates - Pauli X

Pauli Gates - Pauli Y

Pauli Gates - Pauli Z

Hadamard Gate

Phase Gate

RXGate

RYGate

Read/Measurement

Get down to business:

Notation used with Google Cirq

Code example to achieve a specific state |Ψ>

Gates for Multiple Qubits

CNOT Gate

Bell States

Investigating |Θ+> Bell state

Toffoli - CCNOT

CPhase and CZ

Phase KickBack

SWAP

Combining Gates

Chapter 4

Quantum Teleportation

Short fictional story

The Teleportation Process

The Math Behind this Quantum Teleportation

Sending/modulating

Receiving/demodulating

Conclusion

Chapter 5

Deutsch–Jozsa algorithm

Steps of the algorithm

Algorithm examples

Balanced function with 3 Qubits (n=3)

Constant function with 3 Qubits (n=3)

The Math Behind Deutsch–Jozsa algorithm

Scenario when f(x) is a constant function.

Scenario when f(x) is a balanced function.

Conclusion

Chapter 6

Grover algorithm

Algorithm Summary

Understanding the algorithm

Let's start with the problem definition

Additional Definitions

Starting algorithm analysis

Initializing

Oracle phase flip

Grover Diffusion Operator

Conclusion

Chapter 7

Shor Algorithm I

Fundamental mathematics

The structure of our algorithm

The quantum portion of our algorithm

Chapter 8

Quantum Fourier Transform

Discrete Fourier Transform (DFT)

Quantum Fourier Transform (QFT)

Quantum Fourier Transform Circuit

Exemplifying QFT with 1 Qubit

Exemplifying QFT with 2 Qubits

Exemplifying QFT with 4 Qubits

Conclusion

Chapter 9

Quantum Phase Estimation

Concept

Phase Representation

Quantum phase estimation

The quantum phase estimation circuit example

Conclusion

Chapter 10

Shor Algorithm II

From QPE to Order Finding

QPE

Order-finding

Explaining the operator Ux = |gx mod N>

Using the operator Ux = |gx mod N>

Example of Shor's Algorithm: N = 15, g = 7

Building Ux = |7x mod 15>

Analyzing Order Finding for N=15 and g = 7

Applying the invQFT operation

For those curious, where is e²π i j/² ?

Conclusion

Chapter 11

Concluding thoughts

We're off and running, now what?

My Inspiration

It's time to say see you later

Appendix

XOR - ⊕

The notation X ∈ {0,1}n

HAD|X>´s alternative mathematical expression

Sum of the bitwise product

Grover’s Oracle Flip Phase

Number of Grover’s Iterations

GCD Euclidean algorithm

Unitarity of the QFT

References

Chapter 1

Introduction to Quantum Computing

This introduction is inspired by the article written by Scott Aaronson, so I will not write about the history of computers, from the abacus to current computers, but instead starts directly from the core, going to the central point from probabilities (only positive number) to amplitudes (that can be positive, negative until complex number) since quantum mechanics is a highly accurate mathematical model with concepts such as probability and superposition that emerges from observations and experiments of the strange world of atoms and subatomic particles.

Required background

This material assumes familiarity with some foundational mathematical concepts:

Mathematical functions: Understanding how functions operate on inputs to produce outputs.

Trigonometry: Knowledge of sine, cosine, tangent, and their applications in angles and relationships;

Matrices: The ability to work with matrices, including operations like addition, multiplication, and transposition;

Number systems: Comfort with converting between binary, decimal, and hexadecimal representations of numbers;

Complex numbers: An understanding of numbers with both real and imaginary components. Having a grasp of these concepts will be beneficial in following the discussions and code examples presented.

Code examples

The complete source codes for the sample programs can be found in a public Bitbucket repository. While I won't be including the code directly in this book, there are a few reasons for this decision.

Length: The code spans multiple pages, which can disrupt the flow of reading and make it difficult to follow the concepts being explained;

Formatting: Code displays differently in print than in its original programming environment, which can affect readability.

I believe it's more beneficial to focus on the key concepts and functionalities of the programs rather than getting bogged down in the specifics of the code.

However, if you'd like to explore the code in more detail, you can access it through the following location:

Bitbucket Repository: https://bitbucket.org

Source Path:/Nivio/htcfqc/src/main/

These sample programs are implemented in Python, utilizing the Cirq framework within a Jupyter Notebook environment. A basic understanding of Python and Jupyter Notebook will be helpful in following the code.

Cirq

Cirq is an open-source quantum computing framework developed by Google, designed to help developers create, simulate, and run quantum circuits on near-term quantum computers. It provides tools to build and manipulate quantum circuits, optimize them, and simulate their execution on classical computers.

I have chosen it because, as one of the leading frameworks, it aligns perfectly with my other passion, machine learning. My decision was largely influenced by the fact that Google developed TensorFlow Quantum (TFQ), a library that bridges the gap between quantum computing and classical machine learning by integrating TensorFlow's deep learning strengths with Cirq's quantum circuit design and simulation capabilities. Attention, while quantum machine learning isn't one subject of this book, using a framework like Cirq can