Mathematical Perspective on Quantum Monte Carlo Methods

Quantum Monte Carlo (QMC) methods aim at solving the $$N$$ -body quantum problem by means of stochastic algorithms. This chapter provides a pedagogical introduction to the mathematical aspects of the most commonly used QMC methods in electronic structure calculation, namely the variational Monte Carlo (VMC) and the diffusion Monte Carlo (DMC) methods. VMC methods allow one to compute expectation values of the form $$\frac{\langle \psi | \hat{O} |\psi \rangle }{\langle \psi |\psi \rangle }$$ for a given $$N$$ -body wavefunction $$\psi $$, and a given observable $$\hat{O}$$, by means of stochastic sampling algorithms. In particular, VMC methods can be used to compute the energy of $$\psi $$, which reads $$\frac{\langle \psi | H_N |\psi \rangle }{\langle \psi |\psi \rangle }$$, where $$H:N$$ is the $$N$$ -body quantum Hamiltonian of the system. Diffusion Monte Carlo methods consist in rewriting the exact ground state energy of the system, that is the lowest eigenvalue of the Hamiltonian $$H:N$$, as the long-time limit of the expectation value of some stochastic process, and in simulating this stochastic process by particle methods.