Quantum Chemistry and Computing for the Curious: Illustrated with Python and Qiskit® code

✍ Scribed by Keeper L. Sharkey, Alain Chance

• Publisher: Packt Publishing
• Year: 2022
• Tongue: English
• Leaves: 354
• Category: Library

✦ Synopsis

Acquire knowledge of quantum chemistry concepts, the postulates of quantum mechanics, and the foundations of quantum computing, and execute illustrations made with Python code, Qiskit, and open-source quantum chemistry packages

Key Features

• Be at the forefront of a quest for increased accuracy in chemistry applications and computing
• Get familiar with some open source quantum chemistry packages to run your own experiments
• Develop awareness of computational chemistry problems by using postulates of quantum mechanics

Book Description

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

What you will learn

• Understand mathematical properties of the building blocks of matter
• Run through the principles of quantum mechanics with illustrations
• Design quantum gate circuit computations
• Program in open-source chemistry software packages such as Qiskit®
• Execute state-of-the-art-chemistry calculations and simulations
• Run companion Jupyter notebooks on the cloud with just a web browser
• Explain standard approximations in chemical simulations

Who this book is for

Professionals interested in chemistry and computer science at the early stages of learning, or interested in a career of quantum computational chemistry and quantum computing, including advanced high school and college students. Helpful to have high school level chemistry, mathematics (algebra), and programming. An introductory level of understanding Python is sufficient to read the code presented to illustrate quantum chemistry and computing

Table of Contents

1. Introduction
2. Postulates of quantum mechanics
3. Quantum circuit model of computation
4. Molecular Hamiltonians
5. Variational Quantum Eigensolver (VQE) algorithm
6. Beyond Born-Oppenheimer
7. Conclusion
8. References
9. Glossary

✦ Table of Contents

Cover
Title Page
Copyright
Dedication
Foreword
Contributors
Acknowledgments
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
Index
About Packt
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