  Classical non-perturbative simulations of open quantum systems' dynamics face
several scalability problems, namely, exponential scaling of the computational
effort as a function of either the time length of the simulation or the size of
the open system. In this work, we propose the use of the Time Evolving Density
operator with Orthogonal Polynomials Algorithm (TEDOPA) on a quantum computer,
which we term as Quantum TEDOPA (Q-TEDOPA), to simulate non-perturbative
dynamics of open quantum systems linearly coupled to a bosonic environment
(continuous phonon bath). By performing a change of basis of the Hamiltonian,
the TEDOPA yields a chain of harmonic oscillators with only local
nearest-neighbour interactions, making this algorithm suitable for
implementation on quantum devices with limited qubit connectivity such as
superconducting quantum processors. We analyse in detail the implementation of
the TEDOPA on a quantum device and show that exponential scalings of
computational resources can potentially be avoided for time-evolution
simulations of the systems considered in this work. We applied the proposed
method to the simulation of the exciton transport between two light-harvesting
molecules in the regime of moderate coupling strength to a non-Markovian
harmonic oscillator environment on an IBMQ device. Applications of the Q-TEDOPA
span problems which can not be solved by perturbation techniques belonging to
different areas, such as the dynamics of quantum biological systems and
strongly correlated condensed matter systems.
