  Using quantum algorithms to simulate complex physical processes and
correlations in quantum matter has been a major direction of quantum computing
research, towards the promise of a quantum advantage over classical approaches.
In this work we develop a generalized quantum algorithm to simulate any
dynamical process represented by either the operator sum representation or the
Lindblad master equation. We then demonstrate the quantum algorithm by
simulating the dynamics of the Fenna-Matthews-Olson (FMO) complex on the IBM
QASM quantum simulator. This work represents a first demonstration of a quantum
algorithm for open quantum dynamics with a moderately sophisticated dynamical
process involving a realistic biological structure. We discuss the complexity
of the quantum algorithm relative to the classical method for the same purpose,
presenting a decisive query complexity advantage of the quantum approach based
on the unique property of quantum measurement. An accurate yet tractable
quantum algorithm for the description of complex open quantum systems (like the
FMO complex) has a myriad of significant applications from catalytic chemistry
and correlated materials physics to descriptions of hybrid quantum systems.
