# Introduction to Quantum Monte Carlo (QMC) Methods

Quantum Monte Carlo methods use random numbers to numerically solve quantum mechanical problems, hence the name. By the nature of its broad definition, many methods belong to the QMC family. Variational Monte Carlo (VMC) is conceptually the simplest. Other methods include Green’s Function Monte Carlo (GFMC), Diffusion Monte Carlo (DMC), Reptation Monte Carlo (RMC), Coupled-Electron Ion Monte Carlo (CEIMC), Path Integral Monte Carlo (PIMC), World Line Monte Carlo, Auxiliary Field Monte Carlo, Determinantal Monte Carlo and so on.

## VMC

VMC is essentially a trial and error method that exploits the quantum variational principle. That is: any normalized trial wave function necessarily gives an expectation value of the Hamiltonian, aka energy, that is higher than the true ground-state energy. One can therefore parametrize a guess for the ground-state wave functions and judge its quality by evaluating the energy. The wave function with the lowest energy is (assumed to be) the best guess within the parametrization. Stochastic integration is best suited for quickly performing high-dimensional integration such as the one required in estimating the expectation value of the Hamiltonian. In particular, importance sampling is the go-to integrator for VMC.

## DMC

Although fundamentally different, algorithmically, DMC is only one step beyond VMC. The only difference is that in addition to sampling, DMC introduces a growth/death (or an equivalent weighting) procedure to an ensemble of walkers that converge to a stochastic realization of a product of trial and true ground state wave functions. Mathematically, DMC is a stochastic implementation of the power method. It evolves the trial wave function (represented by an ensemble of walkers) in imaginary time using the exact Hamiltonian (in the zero timestep limit) thus samples the trial-true product state (in the infinite time limit).  Of course there’s a cost to be paid for such improvement. That is, the improved wave function will merely exist as a collection of sample points and there is currently no way to recover a compact representation.

Core Concepts: (1,2 -> VMC) (3,4,5 -> DMC)

1. Ritz Variational Principle
2. Monte Carlo Integration
3. The Power Method
4. Imaginary Time Sch$$\ddot{\text{o}}$$dinger Equation
5. Feynman-Kac (PDE and diffusion)

References:

1. [Hammond, Lester, Reynolds] Monte Carlo Methods in Ab Initio Quantum Chemistry, 1994
2. [Ceperley] H and He 2012 Section II.D. and references there in
3. [Foulkes] QMC 2001 Section III
4. [Umrigar] DMC Time-step Error 1993

Devil in the Detail:

1. The Sign Problem
2. Convergence and Statistics Collection
3. Statistical Correlation
4. Cusp Condition
5. Undersampling (Infinite Variance)
6. Timestep Error
1. Use small enough timestep such that this error is below statistical noise.
2. Extrapolate to zero-timestep limit (extrapolation error)
7. Population Control Bias
1. Use big enough population such that this error is below statistical noise.
2. Especially important when trial wavefunction is sub-optimal
8. Mixed-estimator Bias
1. Not really a problem for energy (used in population control)
2. Especially important when trial wavefunction is sub-optimal
3. Use Reptation Monte Carlo or Forward Walking

Projects: