This repository contains the code for the following two papers.

• Transition of Anticoncentration in Gaussian Boson Sampling
• The Second Moment of Hafnians in Gaussian Boson Sampling

The Recursion folder contains the subfolders MathematicaRecursion and JuliaRecursion. MathematicaRecursion contains Mathematica files (these can also be opened and run with WolframScript, which is a free version of Mathematica). The recursion.nb file performs the recursion to exactly compute the values of g(n, a_{12}, a_{13}, a_{23}) (with g(n, 0, 0, 0) being a special case). It loads the precomputed values from computedG.csv, and as it computes more values, it appends the resulting values to computedG.csv.

The JuliaRecursion folder performs the same recursion as is performed in the MathematicaRecursion folder. However, it is written in Julia and so is much much faster. The code is however mostly uncommented. But it is a direct translation of the Mathematica code, and the Mathematica code is heavily commented. Hence, if you want to understand the code, we recommend reading the Mathematica code; if you want to run the code, we recommend running the Julia code.

We emphasize that in both the Mathematica and Julia code, polynomials in k (e.g. a + b k + c k^2) are represented as arrays ({a, b, c}). Multiplying and adding polynomials is then done via functions on the array representations. Using this representation dramatically speeds up the code in comparison to e.g. working with k symbolically in Mathematica.

In the sanityChecks.nb file, we load the computed values of g from Recursion/JuliaRecursion/computedG.csv and perform some of the sanity checks described in the paper on it.

In the plot-fits.nb file, we load the computed values of g from Recursion/JuliaRecursion/computedG.csv and make some plots.

The NumericalApprox folder contains the code to (a) generate numerical estimates of the fourth moment of the Hafnian (second moment in GBS probabilities) (numerical_approx_fourth_moment.py), and to (b) plot those estimates on top of the analytical results we find in the Recursion folder.

I ran the Julia code Recursion/JuliaRecursion/recursion.jl up to n=40. Unfortunately, Recursion/JuliaRecursion/computedG.csv is very large now (about 198 MB), and git push is freezing so I can't upload the full file to GitHub.

The current Recursion/JuliaRecursion/computedG.csv file on GitHub goes only to n=9. The file that contains up to n=40 is on Google Drive.

The easiest way to play around with the expressions is to clone the repo, download the computedG.csv file from Drive and replace the Recursion/JuliaRecursion/computedG.csv file with it, then run Analysis/sanityChecks.nb.