Monte Carlo and Sampling

Monte Carlo methods estimate quantum observables by sampling a representation of a wavefunction, density matrix, or projection process. The useful representation determines the estimator, the update rules, and whether the weights can be treated as probabilities. This chapter explains how to make those choices and check both statistical uncertainty and systematic errors.

Enter this chapter

Begin with Monte Carlo basics and Markov-chain Monte Carlo. The first develops variance and importance sampling; the second explains stationarity, rejection, and mixing. Read autocorrelation and effective sample size before interpreting a production error bar.

Use reweighting and error bars whenever the reported quantity is a weighted ratio or nonlinear function of correlated measurements. This includes energy differences, response functions, and average-sign-corrected observables.

Match the representation to the question

Variational Monte Carlo samples a trial wavefunction and optimizes its energy. Its principal systematic limitation is the trial family. Diffusion Monte Carlo projects in imaginary time using drift, diffusion, and branching; time-step, population, estimator, and fermionic constraint errors must be distinguished.

Path-integral discretization constructs a finite-slice thermal measure. Path-integral Monte Carlo samples its bead configurations, including collective path motion. Worldline Monte Carlo works with occupation histories and must change all relevant winding or permutation sectors. Auxiliary-field Monte Carlo samples fields after tracing fermions into determinants.

The sign-problem page explains when negative or complex weights make such sampling exponentially costly. Positivity is a property of the chosen model and representation; rapid mixing is a further requirement.

A benchmark ladder

A one-dimensional Gaussian tests independent sampling: for $p(x)=e^{-x^2}/\sqrt\pi$, $\langle x^2\rangle=1/2$ and the independent standard error is $1/\sqrt{2N}$. A two-state Metropolis chain then tests rejections and stationary weights. An autoregressive sequence with correlation coefficient $0.9$ tests the variance-inflation factor $19$.

For quantum estimators, take the oscillator trial family $\psi_a(x)\propto e^{-ax^2/2}$ with $a>0$ in units $m=\hbar=\omega=1$. Its energy is

$$E(a)=\frac14\left(a+\frac1a\right),$$

and its exact-width local energy has zero variance. The finite-slice oscillator path integral supplies a separate exact Gaussian covariance at each slice count. Together these tests distinguish sampler errors from variational and discretization errors.

For lattice or fermionic methods, compare small systems with exact diagonalization at the same temperature, particle-number convention, and boundary conditions.

Statistical precision is one part of the answer

A result should specify the ensemble, estimator, update units, comparisons between different starting configurations, independent chains, autocorrelation treatment, and uncertainty. Check the relevant systematic variations separately: time slices, projection time, walker population, occupation cutoff, system size, or trial state.

More samples narrow uncertainty around the algorithm’s current expectation. They do not remove a wrong acceptance ratio, a trapped sector, a fixed-node bias, or an insufficiently fine imaginary-time action. A useful convergence study identifies which changes move the answer and which only reduce its statistical scatter.

References

• Ceperley, David M., “Path integrals in the theory of condensed helium,” Reviews of Modern Physics 67, 279–355 (1995), doi:10.1103/RevModPhys.67.279 — density matrices, path sampling, and estimators.
• Flyvbjerg, H., and H. G. Petersen, “Error estimates on averages of correlated data,” The Journal of Chemical Physics 91, 461–466 (1989), doi:10.1063/1.457480 — blocking correlated measurements.
• Foulkes, W. M. C., Lubos Mitas, R. J. Needs, and G. Rajagopal, “Quantum Monte Carlo simulations of solids,” Reviews of Modern Physics 73, 33–83 (2001), doi:10.1103/RevModPhys.73.33 — variational and diffusion Monte Carlo with error analysis.
• Hastings, W. K., “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika 57, 97–109 (1970), doi:10.1093/biomet/57.1.97 — stationary sampling and proposal correction.
• Troyer, Matthias, and Uwe-Jens Wiese, “Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations,” Physical Review Letters 94, 170201 (2005), arXiv:cond-mat/0408370 — generic sign-problem complexity and its limits.