Gaussian Boson Sampling as a Calibrated Variance-Reduction Engine: A DSFL Explanation

Gaussian Boson Sampling (GBS) generates photon-count patterns whose probabilities are governed by hafnian functionals of an encoded covariance. Recent theory shows that GBS samples can approximate Gaussian expectation problems with provable sample-complexity advantages over Monte Carlo (MC) on an open, sizable subset of instances. We give a unifying explanation via the Deterministic Statistical Feedback Law (DSFL). First, GBS acts as a calibrated, admissible map that intertwines target statistics and physical sampling. Second, tuning the mean photon number is a genuine calibration step that minimizes a DSFL residual. Third, principal-angle (Friedrichs) geometry quantifies conditioning and leakage between statistical and sampling features. Fourth, a Lyapunov envelope provides non-asymptotic error decay and design rules. From these ingredients we derive bounds that recover and sharpen known GBS scalings, predict regimes where GBS outperforms MC, and specify a practical hybrid policy that switches by degree. Experiments with simulated distributions and hardware-aligned kernels support the theory and validate the tuning and selection rules.