The paper by Andrew Lucas explores the application of adiabatic quantum optimization (AQO) to solve NP-complete and NP-hard problems. The key idea is to design a quantum Hamiltonian \( H_P \) whose ground state encodes the solution to a problem, and another Hamiltonian \( H_0 \) whose ground state is easy to find and prepare. By adiabatically transitioning from \( H_0 \) to \( H_P \), the system will remain in the ground state of \( H_P \) if the transition is slow enough. The paper discusses the challenges and potential of AQO, including the complexity of the Hamiltonians required and the experimental progress in building devices capable of running such algorithms. Lucas focuses on Ising spin glasses, which are quantum versions of classical Ising models, as a way to encode NP problems. He reviews and extends the constructions of Ising Hamiltonians for various NP problems, such as number partitioning, graph partitioning, clique problems, binary integer linear programming, covering and packing problems, and coloring problems. Each problem is formulated as an Ising Hamiltonian, with the ground state energy corresponding to the solution of the problem. The paper also addresses the practical considerations for implementing these Hamiltonians on quantum devices, such as the number of spins required, the need for ancilla spins, and the energy scale separations. It highlights the importance of reducing the number of spins and ensuring that the Hamiltonians can be embedded onto the device's graph structure. Finally, the paper discusses the solution to the Hamiltonian cycles problem and its extension to the traveling salesman problem, providing a comprehensive framework for using Ising Hamiltonians to solve a wide range of NP problems.The paper by Andrew Lucas explores the application of adiabatic quantum optimization (AQO) to solve NP-complete and NP-hard problems. The key idea is to design a quantum Hamiltonian \( H_P \) whose ground state encodes the solution to a problem, and another Hamiltonian \( H_0 \) whose ground state is easy to find and prepare. By adiabatically transitioning from \( H_0 \) to \( H_P \), the system will remain in the ground state of \( H_P \) if the transition is slow enough. The paper discusses the challenges and potential of AQO, including the complexity of the Hamiltonians required and the experimental progress in building devices capable of running such algorithms. Lucas focuses on Ising spin glasses, which are quantum versions of classical Ising models, as a way to encode NP problems. He reviews and extends the constructions of Ising Hamiltonians for various NP problems, such as number partitioning, graph partitioning, clique problems, binary integer linear programming, covering and packing problems, and coloring problems. Each problem is formulated as an Ising Hamiltonian, with the ground state energy corresponding to the solution of the problem. The paper also addresses the practical considerations for implementing these Hamiltonians on quantum devices, such as the number of spins required, the need for ancilla spins, and the energy scale separations. It highlights the importance of reducing the number of spins and ensuring that the Hamiltonians can be embedded onto the device's graph structure. Finally, the paper discusses the solution to the Hamiltonian cycles problem and its extension to the traveling salesman problem, providing a comprehensive framework for using Ising Hamiltonians to solve a wide range of NP problems.