Second moment of Hafnians in Gaussian boson sampling

Gaussian boson sampling is a popular method for experimental demonstrations of quantum advantage, but many subtleties remain in fully understanding its theoretical underpinnings. An important component in the theoretical arguments for approximate average-case hardness of sampling is anticoncentration, which is a second-moment property of the output probabilities. In Gaussian boson sampling these are given by hafnians of generalized circular orthogonal ensemble matrices. In a companion work by Ehrenberg et al. [Phys. Rev. Lett. 134, 140601 (2025)10.1103/PhysRevLett.134.140601], we develop a graph-Theoretic method to study these moments and use it to identify a transition in anticoncentration. In this work, we find a recursive expression for the second moment using these graph-Theoretic techniques. While we have not been able to solve this recursion by hand, we are able to solve it numerically exactly, which we do up to Fock sector 2n=80. We further derive analytical results about the second moment. These results allow us to pinpoint the transition in anticoncentration and furthermore yield the expected linear cross-entropy benchmarking score for an ideal (error-free) device.