  Quasiprobability representation is an important tool for analyzing a quantum
system, such as a quantum state or a quantum circuit.
  In this work, we propose classical algorithms specialized for approximating
outcome probabilities of a linear optical circuit using $s$-parameterized
quasiprobability distributions. Notably, we can reduce the negativity bound of
a circuit from exponential to at most polynomial for specific cases by
modulating the shapes of quasiprobability distributions thanks to the
norm-preserving property of a linear optical transformation.
  Consequently, our scheme renders an efficient estimation of outcome
probabilities with precision depending on the classicality of the circuit.
  Surprisingly, when the classicality is high enough, we reach a
polynomial-time estimation algorithm within a multiplicative error.
  Our results provide quantum-inspired algorithms for approximating various
matrix functions beating best-known results. Moreover, we give sufficient
conditions for the classical simulability of Gaussian boson sampling using the
approximating algorithm for any (marginal) outcome probability under the
poly-sparse condition.
  Our study sheds light on the power of linear optics, providing plenty of
quantum-inspired algorithms for problems in computational complexity.
