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

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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