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Two recent landmark experiments have performed Gaussian boson sampling (GBS)with a non-programmable linear interferometer and threshold detectors on up to144 output modes (see Refs.~\onlinecite{zhong_quantum_2020,zhong2021phase}).Here we give classical sampling algorithms with better total variation distanceand Kullback-Leibler divergence than these experiments and a computational costquadratic in the number of modes. Our method samples from a distribution thatapproximates the single-mode and two-mode ideal marginals of the given Gaussianboson sampler, which are calculated efficiently. One implementation sets theparameters of a Boltzmann machine from the calculated marginals using a meanfield solution. This is a 2nd order approximation, with the uniform and thermalapproximations corresponding to the 0th and 1st order, respectively. The $k$thorder approximation reproduces Ursell functions (also known as connectedcorrelations) up to order $k$ with a cost exponential in $k$ and highprecision, while the experiment exhibits higher order Ursell functions withlower precision. This methodology, like other polynomial approximationsintroduced previously, does not apply to random circuit sampling because the$k$th order approximation would simply result in the uniform distribution, incontrast to GBS.

Efficient approximation of experimental Gaussian boson sampling