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We show that current noisy quantum computers are ideal platforms for the simulation of quantum many-body dynamics in generic open systems. We demonstrate this using the IBM Quantum Computer as an experimental platform for confirming the theoretical prediction from Wang et al., [Phys. Rev. Lett. 124, 100604 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.100604] of an emergent hierarchy of relaxation timescales of many-body observables involving different numbers of qubits. Using different protocols, we leverage the intrinsic dissipation of the machine responsible for gate errors, to implement a quantum simulation of generic (i.e., structureless) local dissipative interactions.