They tutorially review the determinantal Quantum Monte Carlo method for fermionic systems, using the Hubbard model as a case study. Starting with the basic ingredients of Monte Carlo simulations for classical systems, they introduce aspects such as importance sampling, sources of errors, and finite-size scaling analyses. We then set up the preliminary steps to prepare for the simulations, showing that they are actually carried out by sampling discrete Hubbard-Stratonovich auxiliary fields. In this process the Green's function emerges as a fundamental tool, since it is used in the updating process, and, at the same time, it is directly related to the quantities probing magnetic, charge, metallic, and superconducting behaviours. They also discuss the as yet unresolved "minus-sign problem", and two ways to stabilize the algorithm at low temperatures.

A self-contained and tutorial presentation of the diffusion Monte Carlo method for determining the ground state energy and wave function of quantum systems is provided. First, the theoretical basis of the method is derived and then a numerical algorithm is formulated. The algorithm is applied to determine the ground state of the harmonic oscillator, the Morse oscillator, the hydrogen atom, and the electronic ground state of the H2 ion and of the H2 molecule.

They present a new technics used to develop a new quantum simulation method which allows to calculate the ground-state expectation values of local observables without any mixed estimates.

Review about fermion nodes problems.

This article discusses the basic properties of the path integral method for continuum fermions, focusing on the restricted path integral (RPIMC) approach.

They review random walks and the quantum Monte Carlo methods used to simulate the ground state of many-body quantum systems, namely variational Monte Carlo and projector Monte Carlo.

This article describes the variational and fixed-node diffusion quantum Monte Carlo methods and how they may be used to calculate the properties of many-electron systems.