Variational Monte Carlo estimators

Let $\psi(s)$ be a trial amplitude in an orthonormal configuration basis. Sampling configurations with probability proportional to its squared magnitude converts the Rayleigh quotient into an average of the local energy. That local quantity includes ratios of connected amplitudes under H and need not lie between the smallest and largest eigenvalues for every individual configuration. Its average, rather than each sample, is the variational energy. For complex amplitudes, a finite local-energy sample may be complex even though the exact Hermitian expectation is real.

The energy expectation is an upper bound to the true ground energy of the specified Hamiltonian for any normalized trial state. Sampling uncertainty can move a numerical estimate below that bound without violating the variational theorem. Optimizing parameters lowers an expectation only within the chosen family and may find a local optimum. An exact eigenstate has constant local energy wherever its amplitude is nonzero. Apparent zero sample variance is not proof of exactness if the chain missed support or unexamined zero-amplitude configurations have nonzero residual.

For a proposed move from s to a new configuration, Metropolis–Hastings includes both the target probability ratio and the reverse-to-forward proposal ratio. The proposal factors cancel only for a symmetric proposal. Detailed balance provides a sufficient stationary-distribution condition, while irreducibility and suitable aperiodicity are needed to reach the intended distribution from a generic start. A chain preserving a particle-number sector cannot discover another sector, which is correct only when that restriction matches the variational problem.

Warmup removes sensitivity to an initial distribution only after actual mixing. High acceptance alone does not guarantee exploration: tiny moves can be almost always accepted and still traverse configuration space slowly. Independent initializations, symmetry-related starting states, and monitored slow observables provide useful diagnostics. Saving only every several steps does not remove the need to assess remaining correlations; it usually discards information that blocking analysis could have used. Sampling a fixed trial state also differs from adapting its parameters, which changes the target distribution.

For a stationary record of a real energy estimator, the variance of a sample mean depends on autocorrelation of the measured observable. For complex trial states, use the real part of the local energy for the real standard error below. Define integrated autocorrelation time as one half plus the sum of normalized positive-lag correlations. Then the effective sample count is approximately the saved count divided by twice that time. This convention makes independent samples have integrated time one half. Some literature uses a convention larger by two, so stating the definition prevents an incorrect error estimate.

Blocking groups consecutive samples and studies when the estimated error stabilizes as blocks exceed the correlation scale. Long tails or rare transitions can demand far more data than energy fluctuations initially suggest. These error estimates concern the expectation under the fixed trial probability, not finite-size bias, ansatz bias, or incomplete optimization. The exact-binomial lab gives an independent-sample reference distribution; its narrow bars do not establish that configurations in a real Markov chain are independent.