# Quantum Computing Book Preview

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This is a preview of the book I am writing. It will cover general topics on quantum computing and some quantum physics.

This is not anywhere near the final draft so it may change.

Quantum computing has the potential to help us understand complex chemical reactions more accurately. However, simulating these reactions using a classical computer is difficult because of the large number of variables involved. In this article, we discuss some recent quantum algorithms that can estimate the phase of a chemical reaction using a plane wave basis with sublinear scaling in the number of basis functions N . The Trotter step size bound O(N log N ) for first quantized plane wave algorithms was introduced by Ryan Babbush, and is currently the best possible gate complexity bound for this setting. Recently, Low et al. showed that a quantum walk on a graph can also be implemented using plane waves with sublinear scaling in N. This algorithm has since been improved by Ian D. Kivlichan, who showed that the Trotter step size can be reduced to O(N ) without significantly compromising accuracy. While these algorithms are promising, they are not yet practical for solving real-world chemistry problems. For example, Babbush showed that the gate complexity of their first quantized plane wave algorithm scales as ̃O(η8/3N 1/3), where η is the number of electrons in the system and N is the number of basis functions. In order to achieve chemical accuracy, many more plane waves than this would be required. However, it is possible that future improvements to these algorithms or to quantum hardware will make them more viable for practical applications.

Quantum simulations are important for understanding the behavior of quantum systems. In particular, there are two main ways to perform quantum simulations: first quantization and second quantization. First quantization is when one uses a fixed basis set, such as plane waves, to represent the state of the system. Second quantization is when one uses Gaussian orbitals to represent the state of the system. There are advantages and disadvantages to each representation. Plane waves are especially well suited for use within first quantization because they have a low gate complexity and they scale well as the number of particles in the system increases. Gaussian orbitals are especially well suited for use within…