To characterize the generic behavior of open quantum many-body systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-$1/2$ system of size $\ell$ with up to n-body Lindblad operators, which are n local in the complexity-theory sense. Without locality ($n=l$), the complex Liouvillian spectrum densely covers a “lemon”-shaped support, in agreement with recent findings [S. Denisov et al., Phys. Rev. Lett. 123, 140403 (2019), L. Sa et al., JPA 53, 305303].However, for local Liouvillians ($n < l$), we find that the spectrum is composed of several dense clusters with random matrix spacing statistics, each featuring a lemon-shaped support wherein all eigenvectors correspond to $n$-body decay modes. This implies a hierarchy of relaxation timescales of n-body observables, which we verify to be robust in the thermodynamic limit. Our findings for n locality generalize immediately to the case of spatial locality, introducing further splitting of timescales due to the additional structure.

To test our theoretical prediction, we perform experiments on the IBM quantum computing platform, designing different "waiting circuits" to inject two body dissipative interactions by two qubit entangling gates. We find excellent agreement with our theory and observe the predicted hierarchy of timescales.

[1] K. Wang, F. Piazza, D. J. Luitz ” Hierarchy of relaxation timescales in local random Liouvillians “ Phys. Rev. Lett. 124, 100604 (2020)

[2] O. E. Sommer, F. Piazza, and D. J. Luitz “Many-body Hierarchy of Dissipative Timescales in a Quantum Computer” arXiv:2011.08853