  We introduce an efficient algorithm based on the quantum noise framework for
simulating open quantum systems on quantum devices. We prove that the open
system dynamics can be simulated by repeatedly applying random unitary gates
acting on the system qubits plus a single ancillary bath qubit representing the
environment. This algorithm represents a notable step forward compared to
current approaches, not only beacause the ancilla overhead remains always
constant regardless of the system size, but also because it provides a
perturbative approximation of the full Lindblad equation to first order in the
environment coupling constants, allowing to reach a better target accuracy with
respect to first order approximation in the time step, thus reducing the total
number of steps. When the perturbative approximation does not hold one can take
smaller time steps and the approach reduces to the solution to first order in
the time step. As a future perspective, this framework easily accomodates
non-Markovian effects by relaxing the reset of the bath qubit prescription.
