Verify scaling laws for measurement-induced energy fluctuations and stall offsets

Verify that the within-run fluctuation width scales as the inverse square root of the number of measurement shots, while the distribution of stall offsets is controlled primarily by the optimizer tolerance and the imaginary-time step.

Background

The paper studies energy distributions generated by repeated quantum imaginary-time-evolution simulations in which parameter updates use finite-shot measurement estimates. The authors find that the observed right-skewed aggregate energy distribution is better explained by an inhomogeneous ensemble of trajectories that stall at different optimization points than by fluctuations around a single quadratic optimum.

This interpretation separates two effects: fluctuations within an individual trajectory, attributed primarily to finite-sampling noise, and differences between trajectories in their stall energies, attributed mainly to the optimizer’s termination behavior and the evolution parameters. The authors formulate a falsifiable scaling prediction distinguishing these contributions, but do not test it in the reported simulations.