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

Research Question and Approach

Boson sampling is a leading candidate for demonstrating quantum computational advantage, but its standard photonic implementation faces significant hardware challenges, such as probabilistic photon generation and optical loss. This has motivated a shift toward deterministic, matter-based platforms (e.g., cold atoms, trapped ions, or superconducting circuits). However, these platforms possess finite-dimensional local Hilbert spaces (dimension d), which introduce a "bunching leakage" error when multiple particles occupy the same mode. The authors aim to determine the minimum number of modes (m) required to suppress this leakage and preserve the #P-hard sampling distribution as a function of particle number (n) and local dimension (d).

To address this, the authors develop a unified Lie-algebraic framework that treats bosonic, fermionic, and spin-based samplers as non-interacting evolution on irreducible representations of compact Lie groups. They analyze the bunching leakage using a Dyson-series expansion, decomposing the many-body leakage operator into a sum of deterministic combinatorial operators weighted by independent random variables. This allows the application of non-commutative concentration inequalities to bound the spectral norm of the leakage operator.

The study proves that in a Gaussian model of the transition matrix, the spectral norm of the leakage operator concentrates at O(sqrt(n)), a substantial improvement over the O(n) worst-case estimate previously assumed. This concentration leads to a near-optimal mode-scaling requirement of m = O(n^(1+2/(d-1))).

For the simplest case of hard-core qubits (d=2), the requirement is m = O(n^3), resolving a long-standing conjecture. For spin-1 representations (d=3), the overhead drops to m = O(n^2), which coincides with the collision-free threshold. Numerical simulations across local dimensions d=2 to 5 confirm that these bounds are tight, with the Haar-ensemble norm matching the derived closed-form expressions to sub-percent accuracy.