Analytic Cancellation of Interference Terms and Closed-Form 1-Mode Marginals in Canonical Boson Sampling

Abstract

Although the $k$-mode marginal distributions of Canonical Boson Sampling (CBS) are known to be computable in polynomial time, the physical mechanism driving this computational efficiency remains mathematically opaque. In this work, we provide a direct, bottom-up physical derivation of the exact 1-mode marginal distribution in CBS, computable in $\mathcal{O}(R^2)$ time, where $R$ is the total number of photons. We explicitly bridge this physical derivation with the mathematical theory of rank-1 matrix permanents, proving that multiphoton interference natively reduces to a symmetric polynomial scaled by a factorial bosonic bunching factor. Crucially, we demonstrate that our recursive combinatorial formulation circumvents the algorithmic overhead of characteristic function methods, entirely bypassing the need for polynomial interpolation or Fourier transforms. Finally, we apply this formula to identify macroscopic signatures of bunching, providing a rigorous, highly scalable metric for distinguishing genuine quantum interference from classical distinguishable-particle models using standard threshold detectors.