#### Overview of this book

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
Preface
Chapter 1: Introducing Quantum Concepts
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Chapter 2: Postulates of Quantum Mechanics
Chapter 3: Quantum Circuit Model of Computation
Chapter 5: Variational Quantum Eigensolver (VQE) Algorithm
Chapter 6: Beyond Born-Oppenheimer
Chapter 7: Conclusion
Chapter 8: References
Chapter 9:Glossary
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Appendix B: Leveraging Jupyter Notebooks on the Cloud

# 5.1. Variational method

We illustrate the variational method through both classical and hybrid-quantum methods. We compare VQE to the variational Monte Carlo method. Further, we also compare the results for VQE to the Quantum Phase Estimation (QPE) algorithm, which is not a variational method.

In this section, we cover the following topics:

• Section 5.1.1, The Rayleigh-Ritz variational theorem
• Section 5.1.2, Variational Monte Carlo methods
• Section 5.1.3, Quantum Phase Estimation (QPE)
• Section 5.1.4, Description of the VQE algorithm

## 5.1.1. The Rayleigh-Ritz variational theorem

The Rayleigh-Ritz variational theorem states that the expectation value of the Hamiltonian of a system with respect to the state of an arbitrary wave function () is always an upper bound to the exact ground state energy of the system it describes:

where generally represents time, spatial, and spin variables. This formula is not assuming any particular...