  We present a quantum algorithm based on the Generalized Quantum Master
Equation (GQME) approach to simulate open quantum system dynamics on noisy
intermediate-scale quantum (NISQ) computers. This approach overcomes the
limitations of the Lindblad equation, which assumes weak system-bath coupling
and Markovity, by providing a rigorous derivation of the equations of motion
for any subset of elements of the reduced density matrix. The memory kernel
resulting from the effect of the remaining degrees of freedom is used as input
to calculate the corresponding non-unitary propagator. We demonstrate how the
Sz.-Nagy dilation theorem can be employed to transform the non-unitary
propagator into a unitary one in a higher-dimensional Hilbert space, which can
then be implemented on quantum circuits of NISQ computers. We validate our
quantum algorithm as applied to the spin-boson benchmark model by analyzing the
impact of the quantum circuit depth on the accuracy of the results when the
subset is limited to the diagonal elements of the reduced density matrix. Our
findings demonstrate that our approach yields reliable results on NISQ IBM
computers.
