  BosonSampling is a well-defined scheme for demonstrating quantum supremacy
with photons in near term. Although relying only on multi-photon interference
in nonadaptive linear-optical networks, it is hard to simulate classically.
Here we study BosonSampling using matrix product states, a powerful method from
quantum many-body physics. This method stores the instantaneous quantum state
during the evolution of photons in the optical quantum circuit, which allows us
to reveal the dynamical features of single photons in BosonSampling devices,
such as entanglement entropy growth. We show the flexiblility of this method by
also applying it to dissipative optical quantum circuits, as well as circuits
with fermionic particles. Our work shows that matrix product states is a
powerful platform to simulate optical quantum circuits. And it is readily
extended to study quantum dynamics in multi-particle quantum walks beyond
BosonSampling.
