Observable-dependent convergence in Monte Carlo wave-function simulations and an adaptive remedy
Description
The Monte Carlo wave-function (MCWF) method estimates observables by averaging over an ensemble of stochastic quantum trajectories. In standard practice, the ensemble size N is fixed before the simulation begins. We demonstrate that this strategy is statistically inadequate as a precision-control strategy: the statistical precision of the result is not a property of N alone, but of N jointly with the observable-specific variance-to-mean-squared ratio σ^2/μ^2, which is unknown prior to simulation. Concretely, on a single resonantly driven two-level atom, three observables require trajectory counts spanning a factor of 16.7 × to achieve the same 1% relative precision—a spread invisible to any fixed-N protocol. The consequence is systematic: fixed-N runs simultaneously under-sample high-variance observables (achieved estimated relative SEM up to 5.8 × the target, with no user warning) and over-sample low-variance ones (wasted compute factors up to 29 × ). The statistical root cause is the law N_opt=σ^2/(ε|μ|)^2: the required ensemble size is a function of per-observable population statistics that cannot be collapsed into a universal trajectory budget. The principled resolution is online inference: estimate σ^2 and |μ| from the trajectories themselves and stop as soon as the target precision is reached. We formalise this as an adaptive stopping criterion based on the relative standard error of the mean (RelSEM < ε), computed via Welford’s algorithm at O(1) overhead per trajectory. While tolerance-based Monte Carlo stopping is not itself new (Section 1), we show empirically that an existing jackknife-based implementation lacks a minimum-trajectory safeguard and can halt after as few as two trajectories with error exceeding 90%; LiQuID’s N_min-guarded Welford criterion avoids this failure mode. Three quantitative benchmarks on analytically tractable systems confirm that the criterion halts within ∼ 13% of the asymptotic optimum and eliminates both failure modes of fixed-N simultaneously.