Polynomial speedup in Torontonian calculation by a scalable recursive algorithm

Evaluating the Torontonian function is a central computational challenge in the simulation of Gaussian Boson Sampling (GBS) with threshold detection. In this work, we propose a recursive algorithm providing a polynomial speedup in the exact calculation of the Torontonian compared to state-of-the-art algorithms. According to our numerical analysis the complexity of the algorithm is proportional to \(N^{1.0691}2^{N/2}\) with \(N\) being the size of the problem. We also show that the recursive algorithm can be scaled up to HPC use cases making feasible the simulation of threshold GBS up to \(35-40\) photon clicks without the needs of large-scale computational capacities.