Polynomial speedup in Torontonian calculation by a scalable recursive algorithm

TL;DR

This paper introduces a recursive algorithm that significantly speeds up the exact computation of the Torontonian function, enabling more efficient simulation of Gaussian Boson Sampling with threshold detection on high-performance computing systems.

Contribution

A novel recursive algorithm that achieves polynomial speedup in calculating the Torontonian, facilitating large-scale GBS simulations without extensive computational resources.

Findings

Complexity proportional to N^{1.0691} 2^{N/2}

Scalable to HPC environments for 35-40 photon clicks

Enables feasible simulation of threshold GBS

Abstract

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.