Near-Optimal Mode Scaling for Finite-Dimensional Boson Sampling via Lie-Algebraic Leakage Bounds

Boson sampling demonstrates quantum advantage through the interference of indistinguishable particles, with output probabilities governed by matrix permanents. Realizing it on deterministic, matter-based platforms requires encoding the bosonic modes in finite-dimensional local Hilbert spaces, which introduces a leakage channel absent in linear optics: multi-particle bunching beyond the local truncation $d$. We develop a unified framework for non-interacting sampling on the irreducible representations of compact Lie groups, in which the transition amplitude is the immanant of a submatrix of the single-particle transition matrix, recovering the permanent in the bosonic case. Within this framework we bound the bunching leakage through a Dyson-series analysis: decomposing the correlated many-body leakage operator into independent random matrices and applying non-commutative concentration inequalities, we prove, in a Gaussian model of the transition matrix, that its spectral norm concentrates at $\tilde{O}(\sqrt{n})$ rather than the $O(n)$ worst-case of prior spin-based emulations; the passage to the physical Haar ensemble is reduced to a single submatrix-comparison input, verified at leading order. Exact numerics across local dimensions $d=2$--$5$ indicate that the bound is tight, the Haar-ensemble norm matching the closed form $\sqrt{d(n-d+1)}$ to sub-percent accuracy. This tightens the required mode number from $m=\Omega(n^4)$ to the near-optimal $m=\tilde{\Omega}(n^{1+2/(d-1)})$; for a spin-1 representation ($d=3$) the overhead falls to $m=\tilde{\Omega}(n^2)$, matching the collision-free threshold. The result is independent of particle statistics and applies across finite-dimensional Lie-symmetric architectures, quantifying the spatial resources needed to preserve sampling hardness.