Estimating outcome probabilities of linear optical circuits and its applications

We propose polynomial-time estimation schemes for outcome probabilities of linear optical circuits by using $s$-parameterized quasiprobability distributions. We exploit the Gaussian factors' interchangeability between the quasiprobability distributions of the input state and the measurement operator, which is an intrinsic property of a linear optical circuit. By performing Monte-Carlo sampling, the precision of the additive error estimation continuously changes as the classicality of the Gaussian input state. Furthermore, we find conditions for efficient multiplicative error estimations of the outcome probabilities by using the sampling from log-concave functions. Our results provide quantum-inspired efficient estimating algorithms for various matrix functions, e.g., permanent and hafnian, within additive or multiplicative errors for a certain class of matrices. For the hafnian, we give an efficient algorithm of $|Haf(R)|^2$ for an $M imes M$ complex symmetric matrix $R$ within an additive error $epsilon(a||R||)^M$, with $a simeq 1.502$.

    Our method can also efficiently approximate any outcome probability with any marginal within an additive error of a Gaussian boson sampling circuit with photon-number detectors under a moderate transmission rate as $eta simeq 0.764$. For Gaussian boson sampling circuits with threshold detectors, our scheme is applicable even for the noiseless case, so we can efficiently simulate those circuits for sparse output distribution.