Stochastic numerical methods for many-body quantum systems

This lecture covers topics such as exact diagonalization, variational methods, neural network quantum states, machine learning, and stochastic optimization for many-body quantum systems. It explores the use of density matrix methods, trajectory methods, and variational ansatzes to study quantum systems. The instructor discusses the framework of Lindblad master equation, numerical methods for density matrices, and neural encoding of operators. Examples include the driven-dissipative quantum Ising model and the variational principle for finding ground states. The lecture concludes with the application of neural networks in approximating density matrices and the efficiency of variational Monte Carlo for open quantum systems.