To realize an operator or a quantum gate, we need to set up the hardware so that it has the appropriate energy landscape, which is called the Hamiltonian. A quantum state will then evolve by following the Schrödinger equation of the given Hamiltonian. In this chapter, we will first study how to solve the Schrödinger equation using matrix mechanics with both diagonal and non-diagonal Hamiltonians. Then we will discuss how a quantum gate can be generated for a given Hamiltonian. We will then review a few important 1-qubit quantum gates. We will also discuss the CNOT gate, which is a 2-qubit entanglement gate, and demonstrate how to use it to create an entanglement state by combining it with other 1-qubit gates.