Alternative sampling for variational quantum Monte Carlo

Expectation values of physical quantities may accurately be obtained by the evaluation of integrals within Many-Body Quantum mechanics, and these multi-dimensional integrals may be estimated using Monte Carlo methods. In a previous publication it has been shown that for the simplest, most commonly applied strategy in continuum Quantum Monte Carlo, the random error in the resulting estimates is not well controlled. At best the Central Limit theorem is valid in its weakest form, and at worst it is invalid and replaced by an alternative Generalised Central Limit theorem and non-Normal random error. In both cases the random error is not controlled. Here we consider a new `residual sampling strategy’ that reintroduces the Central Limit Theorem in its strongest form, and provides full control of the random error in estimates. Estimates of the total energy and the variance of the local energy within Variational Monte Carlo are considered in detail, and the approach presented may be generalised to expectation values of other operators, and to other variants of the Quantum Monte Carlo method.

💡 Research Summary

The paper addresses a fundamental statistical shortcoming of the most widely used sampling strategy in continuum quantum Monte Carlo (QMC), namely the standard sampling that draws configurations R from the probability density |Ψ(R)|² and evaluates the local energy E_L(R)=Ψ⁻¹ĤΨ. While this approach yields unbiased estimates of expectation values, the distribution of the local energy often possesses heavy tails. In particular, when the weight |E_L−E₀|⁻² appears in the integrand, the second moment diverges, causing the Central Limit Theorem (CLT) to fail in its strongest form. Consequently, the sample mean does not converge to a normal distribution; instead a Generalised CLT applies, leading to Lévy‑stable (non‑Gaussian) fluctuations and rendering conventional error bars meaningless.

To overcome this, the authors propose a “residual sampling” scheme that restores the strong CLT and thereby provides full statistical control over the random error. The key idea is to re‑weight the sampled configurations with a carefully chosen function w(E_L)=1/(E_L−E₀)², where E₀ is a pre‑estimated reference energy (often the current estimate of the total energy). This weight suppresses contributions from configurations with extreme local‑energy values, effectively modifying the original probability density P(E_L) to a new density P′(E_L)∝w(E_L)P(E_L). Because the tail of P′ decays as |E_L−E₀|⁻⁴, both the first and second moments become finite irrespective of the original distribution’s pathology.

Mathematically, the estimator is written as a ratio of two weighted sums: \