Fundamentals of Quantum Computing, Ising Hamiltonian and Quantum Evolution in Gate-Based Quantum Computers

We present a series of mathematical and linear algebra connections between quantum circuits and quantum gates on one hand, and general quantum evolution and the Lie-Trotter decomposition on the other. These connections are explained intuitively using visualizations. The mathematical definitions of basic quantum gates are discussed. We also cover the basics of quantum circuits, including modifying empty spots in quantum layers using identity gates and the mathematical aspects of this process. The Ising model was simulated using its Hamiltonian, and the correlations observed during measurement were visualized and explained. The article aimed to connect the theoretical foundations of quantum mathematics with its programming implementations. Since quantum fundamentals and quantum programming are relatively hard to grasp at the beginning, the authors tried to explain the basic concepts in a very intuitive way.