General framework for anticoncentration and linear cross-entropy benchmarking in photonic quantum advantage experiments
Photonic architectures are one of the leading platforms for demonstrating quantum computational advantage, with Boson Sampling and Gaussian Boson Sampling as the primary schemes. Yet, we lack for these photonic primitives a systematic theoretical understanding of linear cross-entropy benchmarking (LXEB), which is a central tool for testing quantum advantage proposals. In this work, we develop a representation-theoretic framework for the classical computation of average LXEB scores and second moments of output probability distributions, covering a range of quantum advantage experiments based on scattering $n$-photon states through $m$-mode Haar-random interferometers. Our methods apply in any regime, including the saturated regime, where the (expected) number of photons is comparable to the number of optical modes. The same second-moment techniques also allow us to prove anticoncentration for traditional Fock-state Boson Sampling in the saturated regime. Interestingly, for Gaussian Boson Sampling second moments are not sufficient to establish a meaningful anticoncentration statement. The technical core of our approach rests on decomposing two copies of the $n$-particle bosonic space $\mathrm{Sym}^n(\mathbb{C}^m)$ into irreducible representations of $\mathrm{U}(m)$. This reduces two-copy Haar averages to computing purities of initial states after partial traces over particles, highlighting the role that particle entanglement plays for LXEB and anticoncentration.