The paper introduces the concept of quantum annealing (QA) as a method to solve optimization problems by incorporating quantum fluctuations into the simulated annealing process. The authors test this idea using the transverse Ising model, where the transverse field, analogous to temperature in classical annealing, controls the rate of transition between states. The goal is to find the ground state of the Hamiltonian with high accuracy as quickly as possible. The time-dependent Schrödinger equation is solved numerically for small systems with various exchange interactions, and the results are compared with classical annealing (SA) methods. The study finds that QA generally leads to the ground state with a higher probability than SA under the same annealing schedule. For the ferromagnetic model, QA shows better convergence to the ground state compared to SA, especially for faster annealing schedules. In the frustrated model, QA also performs better, and in the Sherrington-Kirkpatrick model of spin glasses, QA is more effective than SA. The authors also solve the single-spin problem exactly for QA under different annealing schedules, finding that the ground state is not reached perfectly for all schedules. The asymptotic approach to the ground state is proportional to \(1/t\) for the ferromagnetic model and \(1/\sqrt{t}\) for the single-spin model. The paper concludes by discussing the computational complexity of QA and the need for further research to implement QA in larger optimization problems.The paper introduces the concept of quantum annealing (QA) as a method to solve optimization problems by incorporating quantum fluctuations into the simulated annealing process. The authors test this idea using the transverse Ising model, where the transverse field, analogous to temperature in classical annealing, controls the rate of transition between states. The goal is to find the ground state of the Hamiltonian with high accuracy as quickly as possible. The time-dependent Schrödinger equation is solved numerically for small systems with various exchange interactions, and the results are compared with classical annealing (SA) methods. The study finds that QA generally leads to the ground state with a higher probability than SA under the same annealing schedule. For the ferromagnetic model, QA shows better convergence to the ground state compared to SA, especially for faster annealing schedules. In the frustrated model, QA also performs better, and in the Sherrington-Kirkpatrick model of spin glasses, QA is more effective than SA. The authors also solve the single-spin problem exactly for QA under different annealing schedules, finding that the ground state is not reached perfectly for all schedules. The asymptotic approach to the ground state is proportional to \(1/t\) for the ferromagnetic model and \(1/\sqrt{t}\) for the single-spin model. The paper concludes by discussing the computational complexity of QA and the need for further research to implement QA in larger optimization problems.