Observables and statistics

Exact expectation and variance functions normalize by the supplied state norm, so callers may provide an unnormalized exact vector. Zero-norm and non-finite states are errors. Matrix rows and columns retain the canonical exact-basis order, and Hermitian variance is evaluated from the centered vector norm rather than from a subtraction of two potentially ill-conditioned moments.

WeightedMoments uses normalized non-negative weights and a parallel-moments merge formula. This makes independently accumulated batches equivalent to one serial accumulator without discarding weight information. Complex estimators retain independent real and imaginary moments through ComplexWeightedMoments.

autocorrelation uses Geyer’s initial-positive-sequence estimator with a common 1/N autocovariance normalization, summing adjacent pairs until the first non-positive pair, and reports both integrated autocorrelation time and effective sample size. The reported time is clamped to at least one, so alternating or very short chains never produce a negative effective sample size. r_hat is the classic equal-length, unsplit Gelman-Rubin diagnostic and returns its within- chain W, between-chain B, value, and algorithm identity. The algorithm is part of the result contract and should be recorded with reports.

disorder_average keeps each realization identifier and computes the weighted fixed-realization mean separately from between-realization variance. One realization reports ensemble variance as unavailable, rather than zero. This prevents sampling uncertainty within one realization from being silently combined with finite-disorder ensemble uncertainty. When per-realization sampling variances are supplied, the summary reports the uncertainty of the weighted ensemble mean, sum_r w_r^2 Var_r / (sum_r w_r)^2, considering only positive-weight records.

The exact backend also provides direct Pauli-string expectations, two-site correlations, axis-labelled Pauli magnetization, spin magnetization using S^a = sigma^a/2, and the exact <S_tot^2> observable from an explicit basis and state. Structure-factor results retain their physical axis and raw versus connected convention. A weighted sublattice helper returns per-configuration signed, absolute, and squared values; callers average those components over configurations.