Eigenstate Thermalization Hypothesis in Quantum Alternative Optimization Ansatz

Eigenstate Thermalization Hypothesis (ETH) is the crossover point between quantum mechanics and thermal mechanics. It is used to discuss the propagation of quantum systems to equilibrium states. This hypothesis is also a tool for looking for robust systems resistant to noise and exhibiting unique behaviors. Recently, some groups have reported the experimental detection of ETH. However, the ETH on quantum algorithms has not been studied well. ETH may also be a tool for searching for new and accurate quantum algorithms. Therefore, I surveyed the effect of ETH on the time propagation of states and energies on Quantum Alternative Optimization Ansatz of Traveling Salesman Problem and Graph Coloring Problem. As a result, I revealed that Hamiltonian’s initial states and graph shape are crucial for ETH. The more time the time average of energies takes to drop ETH, the more accurate the calculation becomes.