  Gaussian boson sampling (GBS) allows for a way to demonstrate quantum
supremacy with the relatively modest experimental resources of squeezed light
sources, linear optics, and photon detection. In a realistic experimental
setting, numerous effects can modify the complexity of the sampling, in
particular loss, partial distinguishability of the photons, and the use of
threshold detectors rather than photon counting detectors. In this paper, we
investigate GBS with partial distinguishability using an approach based on
virtual modes and indistinguishability efficiency. We develop a model using
these concepts and derive the probabilities of measuring a specific output
pattern from partially distinguishable and lossy GBS for both types of
detectors. In the case of threshold detectors, the probability as calculated by
the Torontonian is a special case under our framework. By analyzing the
expressions of these probabilities we propose an efficient auxiliary classical
simulation algorithm which can be used to calculate the probabilities. Our
model and the auxiliary algorithm provide foundations for an approximation
method for the probability calculation and simulation algorithms not only
compatible with existing state-of-the-art simulation algorithms for ideal GBS
but also reduce their complexities exponentially depending on the
indistinguishability. Using the approximation method and the simulation
algorithms we show how the boundary of quantum supremacy in GBS can be pushed
further by partial distinguishability.
