  We provide a noisy intermediate-scale quantum framework for simulating the
dynamics of open quantum systems, generalized time evolution, non-linear
differential equations and Gibbs state preparation. Our algorithm does not
require any classical-quantum feedback loop, bypass the barren plateau problem
and does not necessitate any complicated measurements such as the Hadamard
test. We introduce the notion of the hybrid density matrix, which allows us to
disentangle the different steps of our algorithm and delegate classically
demanding tasks to the quantum computer. Our algorithm proceeds in three
disjoint steps. First, we select the ansatz, followed by measuring overlap
matrices on a quantum computer. The final step involves classical
post-processing data from the second step. Our algorithm has potential
applications in solving the Navier-Stokes equation, plasma hydrodynamics,
quantum Boltzmann training, quantum signal processing and linear systems. Our
entire framework is compatible with current experiments and can be implemented
immediately.
