Quantum Chemistry and Computing for the Curious E-Book

Alex Khan

• Herausgeber: Packt Publishing
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 295

Veröffentlichungsjahr: 2022

Beschreibung

Explore quantum chemical concepts and the postulates of quantum mechanics in a modern fashion, with the intent to see how chemistry and computing intertwine. Along the way you’ll relate these concepts to quantum information theory and computation. We build a framework of computational tools that lead you through traditional computational methods and straight to the forefront of exciting opportunities. These opportunities will rely on achieving next-generation accuracy by going further than the standard approximations such as beyond Born-Oppenheimer calculations.
Discover how leveraging quantum chemistry and computing is a key enabler for overcoming major challenges in the broader chemical industry. The skills that you will learn can be utilized to solve new-age business needs that specifically hinge on quantum chemistry

Leseprobe

Quantum Chemistry and Computing for the Curious

Illustrated with Python and Qiskit® code

Keeper L. Sharkey

Alain Chancé

BIRMINGHAM—MUMBAI

Quantum Chemistry and Computing for the Curious

Copyright © 2022 Packt Publishing

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To my mother, Karen, for always encouraging me to pursue my love of chemistry and who unexpectedly passed away during the writing process of this book.

- Keeper L. Sharkey

To Elaine, my wife, this book is dedicated with love.

- Alain Chancé

Foreword

I am honored to write this foreword to Quantum Chemistry and Computing for the Curious.

I met Keeper in 2019 during the Quantum.Tech Congress in Boston, where she was presenting her ideas on quantum chemistry. It was clear that she was the leading expert in this field and stayed in touch with her ever since. I have worked in various areas of quantum computing with mostly financial use cases; however, I have been exposed to some quantum machine learning and quantum chemistry use cases as well. My general sense has been that the literature and examples that lay out how quantum computing applies to chemistry use cases are sorely lacking.

In 2021, during a conversation, I encouraged Keeper to write this book for the benefit of the quantum computing ecosystem, and it turned out she had already been thinking of it and agreed. I was even more excited to hear that Alain, who is well known in the Qiskit community, had agreed to co-author the book with her, knowing that it would add to the useability and make the topic much more approachable to those with some Qiskit background.

If you are reading this, clearly you have an interest in quantum computing, but also in molecule simulations and quantum chemistry use cases. Richard Feynman wrote that "Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy." All the points of this statement are true and beautifully come out in this book.

Most people in quantum computing are aware that we use the Variational Quantum Eigensolver (VQE) to efficiently obtain the potential energy surfaces (PES) for small molecules. However, applying this approach requires understanding classical methods that may or may not be variational, quantum mechanics, developing the energy equation or Hamiltonian of the molecule, making very specific assumptions, such as the Born-Oppenheimer (BO) approximation, and then converting the Hamiltonian into a quantum circuit before it can be solved using VQE. The Hartree-Fock (HF) theory is used to describe the motion of each electron by a molecular orbital, which in turn is made up of a linear set of atom-centered basis functions. STO-3G is one example. Various techniques, including Bravyi-Kitaev (BK) and Jordan-Wigner (JW) are used to transform a chemical Hamiltonian into a qubit Hamiltonian. We need to ensure that these transformations use qubit operators that represent the fermionic operators. We also need to ensure certain symmetries related to the number of electrons, spin, and time-reversal are preserved during this transformation. This is the convergence of many disciplines and where things get complicated very quickly.

Keeper and Alain lead the reader step-by-step through the many topics, techniques, and fundamentals with detailed explanations and code samples. The book introduces quantum concepts such as the structure of light and the atom and then dives into the required areas of quantum mechanics, such as electron orbital structure, the Pauli exclusion principle (PEP), and Schrödinger's equations. Next, the book introduces key quantum computing concepts, including the Bloch sphere, quantum gates, Bell states, the density matrix, and symmetrized versus anti-symmetrized states. We then dive into the core of quantum simulation with the BO approximation, Fock space, creation and annihilation operators, basis sets, and fermionic to qubit mappings, before introducing the reader to Qiskit Nature as a tool to easily apply the above principles. There are detailed explanations, and the equations and concepts are carefully developed and expanded in far more detail than even I would have imagined. In the hero's journey, all our intellectual faculties are challenged as we finally reach our goal as we begin to put all the concepts that we have meticulously mastered through the VQE algorithm and find the ground state of three different molecules on a quantum computer.

In 2019, Keeper mentioned to me that one day, quantum computers could provide more accurate simulations of complex molecules, where approximations required for classical computations would not be needed. Is this the future for quantum simulations? You will find out more as Keeper takes us beyond Born-Oppenheimer.

This book is a goldmine for those wanting the breadth and depth of information required to understand quantum simulations in one place. Keeper and Alain have made a sizable and worthy contribution to the growing wealth of quantum computing literature. I think Richard Feynman would be proud.

Alex Khan

Entrepreneur, advisor, and educator in quantum computing

Baltimore, MD

March 2022

Contributors

About the authors

Keeper L. Sharkey, PhD is the founder and CEO of ODE, L3C, a social enterprise that serves through Quantum Science, Technology and Research, qSTAR. She is Chair of Quantum Applied Chemistry at the Quantum Security Alliance. She obtained a PhD in chemical physics from the University of Arizona as a US National Science Foundation graduate research fellow, May 2015, and a Bachelor of Science in both mathematics and chemistry, May 2010. She remains a Designated Scientific Research Campus Colleague at the University of Arizona. She has published over 30 manuscripts in top peer-reviewed journals regarding non-Born-Oppenheimer quantum mechanical finite-nuclear mass variational algorithms and has been cited over 400 times; H-index and i10-index of 10.

Alain Chancé is business advisor to ODE, L3C and is the founder and CEO of Quantalain SASU, a management consulting startup. He has over 30 years of experience in major enterprise transformation projects with a focus on data management and governance gained in major management consulting firms. He has a diploma ingénieur civil des Mines from École des Mines de Saint-Étienne (1981).

He is a Qiskit® Advocate and is an IBM Certified Associate Developer - Quantum Computation using Qiskit® v0.2X since 2021. He has completed a number of hackathons pertaining to quantum computing since 2018.

Acknowledgments

We would like to thank the technical reviewer, Bruno Fedrici, PhD, Professor Ludwik Adamowicz for insights into quantum education, Robert and Suzanne Scifo for artistic renderings of physics and chemistry concepts as it relates to historical quotes in the chapters, Mellissa Larson for photographing the lead author, and Quantum Interns of ODE, L3C, who proofread the chapters and assisted in compiling the glossary: Bhagya Gopakumar and Sneha Thomas.

About the reviewer

Bruno Fedrici has a PhD in quantum engineering from the University of Nice Sophia Antipolis along with a university certificate in digital transformation from the University of Lyon. He contributes to the public and business awareness of quantum technologies by providing a bridge between higher education, research, and industry. For three years now, he has been introducing quantum computing basics and quantum-safe security solutions to executives and technical leaders as well as to computer science and engineering students.

Bruno is a lecturer in quantum information science at INSA Lyon. He has also launched Quantum for Everyone, a new online course for non-technical business professionals. Currently Bruno is also a program manager at Quantum Business Europe, a new event focusing on end user applications of quantum technologies.

Table of Contents

Preface

Chapter 1: Introducing Quantum Concepts

Technical requirements

1.1. Understanding the history of quantum chemistry and mechanics

1.2. Particles and matter

Elementary particles

Composite particles

1.3. Quantum numbers and quantization of matter

Electrons in an atom

The wave function and the PEP

1.4. Light and energy

Planck constant and relation

The de Broglie wavelength

Heisenberg uncertainty principle

Energy levels of atoms and molecules

Hydrogen spectrum

Rydberg constant and formula

Electron configuration

Schrödinger's equation

Probability density plots of the wave functions of the electron in a hydrogen atom

1.5. A brief history of quantum computation

1.6. Complexity theory insights

Summary

Questions

Answers

References

Chapter 2: Postulates of Quantum Mechanics

Technical requirements

2.1. Postulate 1 – Wave functions

2.1.1. Spherical harmonic functions

2.1.2. Addition of momenta using CG coefficients

2.1.3. General formulation of the Pauli exclusion principle

2.2. Postulate 2 – Probability amplitude

2.2.1. Computing the radial wave functions

2.2.2. Probability amplitude for a hydrogen anion

2.3. Postulate 3 – Measurable quantities and operators

2.3.1. Hermitian operator

2.3.2. Unitary operator

2.3.3. Density matrix and mixed quantum states

2.3.4. Position operation

2.3.5. Momentum operation

2.3.6. Kinetic energy operation

2.3.7. Potential energy operation

2.3.8. Total energy operation

2.4. Postulate 4 – Time-independent stationary states

2.5. Postulate 5 – Time evolution dynamics

Questions

Answers

References

Chapter 3: Quantum Circuit Model of Computation

Technical requirements

Installing NumPy, Qiskit, QuTiP, and importing various modules

3.1. Qubits, entanglement, Bloch sphere, Pauli matrices

3.1.1. Qubits

3.1.2. Tensor ordering of qubits

3.1.3. Quantum entanglement

3.1.4. Bloch sphere

3.1.5. Displaying the Bloch vector corresponding to a state vector

3.1.6. Pauli matrices

3.2. Quantum gates

3.2.1. Single-qubit quantum gates

3.2.2. Two-qubit quantum gates

3.2.3. Three-qubit quantum gates

3.2.4. Serially wired gates and parallel quantum gates

3.2.5. Creation of a Bell state

3.2.6. Parallel Hadamard gates

3.3. Computation-driven interference

3.3.1. Quantum computation process

3.3.2. Simulating interferometric sensing of a quantum superposition of left- and right-handed enantiomer states

3.4. Preparing a permutation symmetric or antisymmetric state

3.4.1. Creating random states

3.4.2. Creating a quantum circuit and initializing qubits

3.4.3. Creating a circuit that swaps two qubits with a controlled swap gate

3.4.4. Post selecting the control qubit until the desired state is obtained

3.4.5. Examples of final symmetrized and antisymmetrized states

References

Chapter 4: Molecular Hamiltonians

Technical requirements

Installing NumPy, Qiskit, and importing the various modules

4.1. Born-Oppenheimer approximation

4.2. Fock space

4.3. Fermionic creation and annihilation operators

4.3.1. Fermion creation operator

4.3.2. Fermion annihilation operator

4.4. Molecular Hamiltonian in the Hartree-Fock orbitals basis

4.5. Basis sets

4.5.1. Slater-type orbitals

4.5.2. Gaussian-type orbitals

4.6. Constructing a fermionic Hamiltonian with Qiskit Nature

4.6.1. Constructing a fermionic Hamiltonian operator of the hydrogen molecule

4.6.2. Constructing a fermionic Hamiltonian operator of the lithium hydride molecule

4.7. Fermion to qubit mappings

4.7.1. Qubit creation and annihilation operators

4.7.2. Jordan-Wigner transformation

4.7.3. Parity transformation

4.7.4. Bravyi-Kitaev transformation

4.8. Constructing a qubit Hamiltonian operator with Qiskit Nature

4.8.1. Constructing a qubit Hamiltonian operator of the hydrogen molecule

4.8.2. Constructing a qubit Hamiltonian operator of the lithium hydride molecule

Summary

Questions

References

Chapter 5: Variational Quantum Eigensolver (VQE) Algorithm

Technical requirements

Installing NumPy, Qiskit, QuTiP, and importing various modules

5.1. Variational method

5.1.1. The Rayleigh-Ritz variational theorem

5.1.2. Variational Monte Carlo methods

5.1.3. Quantum Phase Estimation (QPE)

5.1.4. Description of the VQE algorithm

5.2. Example chemical calculations

5.2.1. Hydrogen molecule (H2)

5.2.2. Lithium hydride molecule

5.2.3. Macro molecule

Summary

Questions

Answers

References

Chapter 6: Beyond Born-Oppenheimer

Technical requirements

Installing NumPy, SimPy, and math modules

6.1. Non-Born-Oppenheimer molecular Hamiltonian

Internal Hamiltonian operator

Explicitly correlated all-particle Gaussian functions

Energy minimization

6.2. Vibrational frequency analysis calculations

Modeling the vibrational-rotational levels of a diatomic molecule

Computing all vibrational-rotational levels of a molecule

6.3. Vibrational spectra for ortho-para isomerization of hydrogen molecules

Summary

Questions

Answers

References

Chapter 7: Conclusion

7.1. Quantum computing

7.2. Quantum chemistry

References

Chapter 8: References

Chapter 9:Glossary

Appendix A: Readying Mathematical Concepts

Technical requirements

Installing NumPy, SimPy, and Qiskit and importing various modules

Notations used

Mathematical definitions

Pauli exclusion principle (PEP) #

Angular momentum quantum number #

Occupation number operator #

Quantum Phase Estimation (QPE) #

Complex numbers

Vector space

Linear operators

Matrices

Eigenvalues and eigenvectors

Vector and matrix transpose, conjugate, and conjugate transpose

Dirac's notation #

Inner product of two vectors

Norm of a vector

Hilbert space

Matrix multiplication with a vector

Matrix addition

Matrix multiplication

Matrix inverse

Tensor product

Kronecker product or tensor product of matrices or vectors

Kronecker sum

Outer product

Hermitian operator

Unitary operator

Density matrix #

Pauli matrices

Anti-commutator #

Anti-commutation #

Commutator

Total wave function #

References

Appendix B: Leveraging Jupyter Notebooks on the Cloud

Jupyter Notebook

Google Colaboratory

IBM Quantum Lab

Companion Jupyter notebooks

References

Appendix C: Trademarks

Other Books You May Enjoy

Preface

"Learning is finding out what you already know. Doing is demonstrating that you know it. Teaching is reminding others that they know just as well as you. You are all learners, doers, teachers."

– Richard Bach

Figure 0.1 – Learning quantum computing and quantum chemistry [authors]

This book aims to demystify quantum chemistry and computing, discuss the future of quantum technologies based on current limitations, demonstrate the usefulness and shortcomings of the current implementations of quantum theory, and share our love of the topic.

This book is not a traditional presentation of quantum chemistry nor quantum computing, but rather an explanation of how the two topics intertwine through the illustration of the postulates of quantum mechanics, particularly with Python code and open-source quantum chemistry packages.

Quantum chemistry has many applications in industry, from pharmaceutical design to energy creation and the development of quantum computing in recent years. With adequate knowledge of quantum chemistry and the postulates of quantum mechanics, we can overcome some of the major hurdles humanity faces and achieve positive impacts. We hope that you can learn sufficient details to be a part of the new and productive solutions moving forward.

Readers we target

All kinds of readers are welcome. However, the people who will benefit the most are those interested in chemistry and computer science at the early stages of learning; advanced high school and early college students, or professionals wanting to acquire a background in quantum chemistry as it relates to computing, both from an algorithm and hardware standpoint. We also summarize useful mathematics and calculus as it relates to solving chemistry problems. The topics will appeal to people of various industry verticals who are interested in a career in quantum computational chemistry and computing.

You will be at the forefront of exciting state-of-the-art opportunities to expand your ideas and start experimenting with your simulations.

A fast path to using quantum chemistry

We chose to write this book in such a way as to demystify the fundamentals of quantum concepts for a curious audience. This book introduces the basics of quantum chemical concepts by describing the five postulates of quantum mechanics, including how these concepts relate to quantum information theory, including basic programming examples of atomic and molecular systems with Python, SimPy [Simpy], QuTiP [QuTiP], and open-source quantum chemistry packages PySCF [PySCF], ASE [ASE_0], PyQMC [PyQMC], Psi4 [Psi4_0], and Qiskit [Qiskit] code. An introductory level of understanding Python is sufficient to read the code, and a browser is all that is required to access the Google Colaboratory and run the companion Jupyter notebooks we provide in the cloud. Each chapter includes an artistic rendering of quantum concepts related to historical quotes.

Through the 1990s, 2000s, and 2010s, there has been amazing progress in the development of computational chemistry packages and, most recently, Qiskit Nature [Qiskit_Nature] [Qiskit_Nat_0]. We outline and introduce basic quantum chemical concepts that are discussed in a modern fashion and relate these concepts to quantum information theory and computation. We use Python, PySCF, and Qiskit Nature for illustrative purposes.

Quantum chemistry

The fundamentals of quantum mechanics and the five postulates directly impact material research and computational chemistry for finding new drugs and catalysts, enabling efficient and cleaner processes for converting chemicals from one form to another. Quantum chemistry is also essential for designing future quantum computers that use the properties of atoms and/or ions. However, quantum chemistry remains an elusive topic that seemingly takes many years to master.

We think that the traditionally long-term achievement of literacy of the topic is directly related to the perceived complexity of the topic and historical approximations made to increase accessibility and usability with conventional computing. With approximation in place and wide acceptance by the scientific community as the only way forward, some fundamental concepts are often overlooked, misunderstood, and eliminated from the disciplines depending on these ideas. We see this as an opportunity to share our love of quantum chemistry in its full potential to enhance the friendliness of and approachability of the topic.

We will share sufficient details so that you understand the limitations that were historically established. For instance, we present a general formulation of the Pauli exclusion principle for all elementary particles that also holds for composite particles, which many textbooks do not adequately explain.

There is more to the quantum story but too much to be included as a first book for the curious. Therefore, we plan to write a following book that expands cutting-edge quantum ideas that are not yet widely used in the scientific community.

How to navigate the book

We advise you to follow the sequential ordering of chapters and gradually master the concepts, methods, and tools that will be useful later in the book.

Chapter 1, Introducing Quantum Concepts, presents a history of quantum chemistry and quantum computing, and introduces the fundamental building blocks of nature, particles and matter, light and energy, and quantum numbers.Chapter 2, Postulates of Quantum Mechanics, gives a non-expert in quantum physics the concepts, definitions, and notation of quantum mechanics and quantum information theory necessary to grasp the content of this book.Chapter 3, Quantum Circuit Model of Computation, introduces the quantum circuit model of computation and Qiskit Nature, an open-source framework that provides tools for computing ground state energy, excited states, and dipole moments of molecules.Chapter 4, Molecular Hamiltonians, presents the molecular Hamiltonian, modeling the electronic structure of a molecule and fermions to qubit mappings.Chapter 5, Variational Quantum Eigensolver (VQE) Algorithm, shows a process for solving the ground state of a molecule, focusing on the Hydrogen molecule, illustrated with the Variational Quantum Eigensolver (VQE) algorithm using Qiskit Nature.Chapter 6, Beyond Born-Oppenheimer, gives a glimpse of the beyond Born-Oppenheimer approaches that have not yet been popularized.Chapter 7, Conclusion, is the opening to the next book.Chapter 8, References, provides a consolidated list of all the references given at the end of each chapter.Chapter 9, Glossary, provides a convenient way to look up terms.Appendix A, Readying Mathematical Concepts, introduces concepts with illustrations in Python code.Appendix B, Leveraging Jupyter Notebooks in the Cloud, explains how to use free environments on the cloud to run the companion Jupyter notebooks we provide.Appendix C, Trademarks, lists all the trademarks of the products used in this book.

To get the most out of this book

With the following software and hardware list you can access the Google Colaboratory (Colab), which is a free Jupyter Notebook environment that runs entirely in the cloud and provides online, shared instances of Jupyter notebooks without having to download or install any software:

Download the example code files

You can download the example code files for this book from GitHub at https://github.com/PacktPublishing/Quantum-Chemistry-and-Computing-for-the-Curious. If there's an update to the code, it will be updated in the GitHub repository.

To download the full version of the companion notebooks you can scan the following QR code or go to the provided link to download them.

https://account.packtpub.com/getfile/9781803243900/code

We also have other code bundles from our rich catalog of books and videos available at https://github.com/PacktPublishing/. Check them out!

Conventions used

There are a number of text conventions used throughout this book.

Code in text: Indicates code words in text, database table names, folder names, filenames, file extensions, pathnames, dummy URLs, user input, and Twitter handles. Here is an example: "There is no loop in a quantum circuit, but we can have a classical loop that appends a quantum sub-circuit. In Qiskit we use the QuantumRegister class to create a register of qubits and the QuantumCircuit class to create a quantum circuit."

A block of code is set as follows:

q = QuantumRegister(2)

qc = QuantumCircuit(q)

qc.h(q[0])

qc.cx(q[0], q[1])

qc.draw(output='mpl')

Any command-line input or output is written as follows:

Mo: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s² 4d⁴

Get in touch

Feedback from our readers is always welcome.

General feedback: If you have questions about any aspect of this book, email us at [email protected] and mention the book title in the subject of your message.

Errata: Although we have taken every care to ensure the accuracy of our content, mistakes do happen. If you have found a mistake in this book, we would be grateful if you would report this to us. Please visit www.packtpub.com/support/errata and fill in the form.

Piracy: If you come across any illegal copies of our works in any form on the internet, we would be grateful if you would provide us with the location address or website name. Please contact us at [email protected] with a link to the material.

If you are interested in becoming an author: If there is a topic that you have expertise in and you are interested in either writing or contributing to a book, please visit authors.packtpub.com.

References

[ASE_0] Atomic Simulation Environment (ASE), https://wiki.fysik.dtu.dk/ase/index.html

[NumPy] NumPy: the absolute basics for beginners, https://numpy.org/doc/stable/user/absolute_beginners.html

[Psi4_0] Psi4 manual master index, https://psicode.org/psi4manual/master/index.html

[PyQMC] PyQMC, a python module that implements real-space quantum Monte Carlo techniques, https://github.com/WagnerGroup/pyqmc

[PySCF] The Python-based Simulations of Chemistry Framework (PySCF), https://pyscf.org/

[Qiskit] Qiskit, https://qiskit.org/

[Qiskit_Nat_0] Qiskit_Nature, https://github.com/Qiskit/qiskit-nature/blob/main/README.md

[Qiskit_Nature] Introducing Qiskit Nature, Qiskit, Medium, April 6, 2021, https://medium.com/qiskit/introducing-qiskit-nature-cb9e588bb004

[QuTiP] QuTiP, Plotting on the Bloch Sphere, https://qutip.org/docs/latest/guide/guide-bloch.html

[Simpy] SimPy Discrete event simulation for Python, https://simpy.readthedocs.io/en/latest

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Chapter 1: Introducing Quantum Concepts

"There are children playing in the streets who could solve some of my top problems in physics, because they have modes of sensory perception that I lost long ago."

– Robert J. Oppenheimer

Figure 1.1 – Girl looking at the image of an atom [Adapted from image licensed by Getty]

Predicting the behavior of matter, materials, and substances not yet measured experimentally is an exciting prospect. Modern computational tools enable you to conduct virtual experiments on freely available resources. Understanding modern models of how chemistry works is essential if you want to get results that match the way Nature operates.

Classical physics works fine for predicting the trajectory of a ball with Newton's law of gravitation or the trajectory of planets around the sun. However, a more accurate description of Nature, especially chemistry, can be found via quantum physics, which encapsulates the postulates of quantum mechanics, the foundation of quantum chemistry, and quantum computing. To gain the next level of understanding of chemistry predictions, quantum chemistry algorithms need to be designed to achieve a high level of accuracy. It is not enough to simply program approximate methods and have a quantum computer run them to achieve higher accuracy than the same method implemented on a classical computer.

The postulates of quantum physics are not considered laws of nature and cannot be proven either mathematically or experimentally; rather, they are simply guidelines for the behavior of particles and matter. Even though it took a few decades for these postulates to be formulated and a century for them to be understood by the broader scientific community, they remain a powerful tool for predicting the properties of matter and particles and are the foundation of quantum chemistry and computing.

This chapter is not an exhaustive presentation of the entire history of quantum physics; however, we will mention some of the crucial figures and introduce the topics that we think are the most influential in the 20th century. We discuss the fundamental concepts of particles and the composition of matter, the physical properties of light and its behavior, plus energy and its relation to matter. We extend these concepts to present the quantum numbers related to certain types of chemical applications and properties that can be specifically used for the advancement of quantum computing and predicting states of matter.

In this chapter, we will cover the following topics:

Section 1.1, Understanding the history of quantum chemistry and mechanicsSection 1.2, Particles and matterSection 1.3, Quantum numbers and quantization of matterSection 1.4, Light and energySection 1.5, A brief history of quantum computationSection 1.6, Complexity theory insights

Technical requirements

A companion Jupyter notebook for this chapter can be downloaded from GitHub at https://github.com/PacktPublishing/Quantum-Chemistry-and-Computing-for-the-Curious, which has been tested in the Google Colab environment, which is free and runs entirely in the cloud, and in the IBM Quantum Lab environment. Please refer to Appendix B – Leveraging Jupyter Notebooks in the Cloud, for more information.

1.1. Understanding the history of quantum chemistry and mechanics

Knowing the development of quantum chemistry during the early part of the 20th century is important in order to understand how the postulates of quantum mechanics were discovered. It will also help you grasp the major approximations that have enabled us to achieve scientific milestones. We will mention concepts that will be discussed and described in later chapters of the book, so don't be concerned if you don't understand what the ideas mean or imply. We want to simply start using the terminology of quantum concepts to give some context to the five postulates of quantum mechanics presented in the rest of the book.

Figure 1.2 – Robert J. Oppenheimer – Ed Westcott (U.S. Government photographer), Public domain, via Wikimedia Commons

Quantum mechanics has been a disruptive topic of conversation in the scientific community for just over a century. The most notable controversy of quantum mechanics is that it gave rise to the atomic bomb during World War II. Robert J. Oppenheimer (Figure 1.2), considered the father of the atomic bomb, is also the inventor of one of the most widely used and influential approximation to date: the Born-Oppenheimer (BO) approximation of 1926 [Intro_BOA_1] [Intro_BOA_2]. This will be described in depth in Chapter 6, Beyond Born-Oppenheimer. The BO approximation assumes that the motions of the nuclei are uncoupled from the motions of the electrons and led to the formulation of the majority of the computational techniques and software packages available to date, including the basic design of a qubit used for quantum computing.

By the time Oppenheimer published his PhD thesis on the BO approximation with Max Born, his academic advisor, many scientists had contributed to quantum chemistry. The term quantum mechanics appeared for the first time in Born's 1924 paper Zur Quantenmechanik [Born]. Quantum mechanics was formulated between 1925 and 1926 with other major contributions from the following:

Max Planck for the Planck constant and the Planck relation (Section 1.4, Light and energy)Louis de Broglie for the de Broglie wavelength (Section 1.3, Quantum numbers and quantization of matter)Werner Heisenberg for the Heisenberg uncertainty principle (Section 1.4, Light and energy)Erwin Schrödinger for the Schrödinger equation (Section 1.4, Light and energy)Paul Dirac for the Dirac equation, a relativistic wave equation for fermionic systems, and for the Dirac notation, also known as bra-ket notation (Section 1.3, Quantum numbers and quantization of matter)Wolfgang Pauli for the Pauli exclusion principle (Section 1.3, Quantum numbers and quantization of matter)

These scientists attended the 5th Solvay conference on quantum mechanics (Figure 1.3) along with other very influential scientists that are not discussed. This image captures the first cohort of quantum scientists that had a great influence on the 20th century.

Figure 1.3 – Solvay Conference on quantum mechanics, 1927. Image is in the public domain

The BO approximation was a necessary development primarily because of the Pauli exclusion principle (PEP), which was formulated in 1925. Pauli described the PEP for electrons, and it states that it is impossible for two electrons of the same atom to simultaneously have the same values of the following four quantum numbers: , the principal quantum number; , the angular momentum quantum number; , the magnetic quantum number; and , the spin quantum number. His work has been further extended to bosonic particles. The PEP leads to a certain type of computational complexity that initiated the necessity for the BO approximation; see Section 1.6, Complexity theory insights for more details. We will go into the detail of quantum quantities and describe PEP for different particle types in Section 1.3, Quantum numbers and quantization of matter.

The rapid development by the aforementioned group of thought leaders came about with the important groundwork laid out by their predecessors and their discoveries related to the hydrogen atom – the simplest of all elements of the periodic table:

Johan Balmer in 1885 discovered the Balmer emission line series [Balmer_series].Johannes Rydberg in 1888 generalized the Balmer equation for all transitions of hydrogen [Chem_spectr].Theodore Lyman from 1906 to 1914 discovered the Lyman series of hydrogen atom spectral lines in the ultraviolet [Lyman_series].Friedrich Paschen in 1908 discovered the Paschen spectral lines in the infrared band [Chem_spectr].

The structure of the hydrogen atom will be discussed in Section 1.4, Light and energy, and outlined computationally in Chapter 5, Variational Quantum Eigensolver (VQE) Algorithm.

The work of Johannes Rydberg led to the definition of the fundamental constant used in spectroscopy. Rydberg worked side-by-side with Walter Ritz in 1908 to develop the Rydberg-Ritz combination principle about the relationship between frequencies and the spectral lines of elements [Rydberg-Ritz]. Rydberg states of atoms are used in quantum computation, and this is discussed in Chapter 3, Quantum Circuit Model of Computation.

A year after the development of the Rydberg-Ritz combination principle, Ritz developed a way to solve the eigenvalue problem [Rayleigh–Ritz], which is widely used today in the field of computational chemistry and is known as the Rayleigh-Ritz variational theorem. This method is the inspiration for the Variational Quantum Eigensolver (VQE) discussed in detail in Chapter 5, Variational Quantum Eigensolver (VQE) Algorithm.

In conjunction with the testing of the Rayleigh-Ritz variational method mechanically by John William Strutt, 3rd Baron Rayleigh, known for the Rayleigh scattering of light, this method is called the Rayleigh-Ritz method even though it was written and formulated by Ritz. In short, it allows for the approximation of the solutions to the eigenvalue problem. His work led to the method of applying the superposition principle to approximate the total wave function; this mathematical expansion is one of the postulates of quantum mechanics described in Chapter 2, Postulates of Quantum Mechanics.

With a better understanding of the hydrogen atom, in 1913, Niels Bohr attempted to describe the structure of atoms in more detail with fundamental concepts about quantization and quantum theory [Bohr_1] [Bohr_2]. He received the Nobel Prize in 1922 for his Bohr model. In his dissertation, many articles verify, predict, and assess very accurate Rydberg states of small atoms (Z < 7) as well as the rotational-vibrational (rovibrational) states of small molecules. Bohr's atomic model describes the electron energy transitions starting from the second, first, and third atom electron layers, known as Balmer series, Lyman series, and Paschen series, and the corresponding hydrogen emission spectrums discovered previously.

In the 1930s, Linus Pauling and Edgar Bright Wilson Jr. popularized quantum mechanics as it is currently applied to chemistry [Pauling]. Pauling eventually received a Nobel Prize for Chemistry in 1954, and later on, he received a Nobel Peace Prize in 1964 for his political activism regarding quantum mechanics.

The development of the postulates of quantum mechanics since these major contributions has remained, in general, the same to date.

Thanks to the development of classical computers and clever computational methods, many computational chemistry packages have been produced to further our understanding of chemistry. Some notable methods, other than the Rayleigh-Ritz variational theorem, are the Quantum Monte Carlo (QMC) [QMC], Hartree-Fock (HF) method, Coupled-cluster (CC), and Density-functional theory (DFT), among others. In this book, we will illustrate some of these methods with Python and open-source quantum chemistry packages such as PySCF, ASE, PyQMC, Psi4, and Qiskit in subsequent chapters.

In the late 20th century, Richard Feynman stated that quantum concepts could be used for quantum computing [Preskill_40y]. Physicists Jonathan Dowling and Gerard Milburn wrote in 2002 that we have entered a second quantum revolution, actively employing quantum mechanics in quantum information, quantum sensing, quantum communication, and analog quantum simulation [Dowling]. We will summarize the history of quantum computation in Section 1.5, A brief history of quantum computation. This second quantum revolution is seen as a way to overcome computational complexity using matter and the postulates of quantum mechanics.

The question becomes: what is the purpose of implementing approximate methods in a quantum computer? Are quantum computers supposed to help us in reaching beyond the aforementioned methods? We intend to address these questions in this book, specifically in Chapter 6, Beyond Born-Oppenheimer.

1.2. Particles and matter

In general, particles and matter have three unique properties that do not change: mass, charge, and magnetic spin. For some particles, these properties can have a value of zero; otherwise, these properties are real numbers and can be measured experimentally. Mass can only be positive, while charge can be positive or negative.

In the following subsections, we will review elementary and composite particles, which include both fermions and bosons. Understanding these kinds of particles is fundamental to the understanding of quantum chemistry and the potential use of quantum computing.

Elementary particles

Elementary particles are either fermions or bosons [Part_1]. The term fermion was coined by Dirac, who was inspired by the physicist Enrico Fermi. Elementary boson particles are part of the Standard Model [Std_model] and do not necessarily take part in quantum chemistry, but rather fundamental physics.

The electron () is the primary elementary fermionic particle associated with quantum chemistry. Electrons have a mass of 9.1093837015 x 10-31 kilograms (kg) [e_mass] and an electric charge of negative one (-1). The size of the electron is on the order of approximately 10-15 centimeters (cm). In most computational methods simulations, we change the reference mass so that an electron's mass is equal to 1, making the computation easier. There are also the muon () and tau () particles, which have a negative one electric charge (-1) but are much heavier than the electron. The associated antiparticles, the positron (), antimuon (), and antitau (