Speeding up the classical simulation of Gaussian boson sampling with limited connectivity

Gaussian boson sampling (GBS) plays a crucially important role in demonstrating quantum advantage. As a major imperfection, the limited connectivity of the linear optical network weakens the quantum advantage result in recent experiments. In this work, we introduce an enhanced classical algorithm for simulating GBS processes with limited connectivity. It computes the loop Hafnian of an \(n \times n\) $n\times n$ symmetric matrix with bandwidth w in \(O(nw2^w)\) $O(nw{2}^w)$ time. It is better than the previous fastest algorithm which runs in \(O(nw^2 2^w)\) $O(n{w}^2{2}^w)$ time. This classical algorithm is helpful on clarifying how limited connectivity affects the computational complexity of GBS and tightening the boundary for achieving quantum advantage in the GBS problem.