For a 1D shift-invariant nearest-neighbor quantum chain, deciding the long-term average single-site state is RE-complete with a single defect, and EXPSPACE- or PSPACE-complete for finite lattices with i.i.d. inputs.
Cubitt, David Perez-Garcia and Michael M
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More on Intractability of Thermalization: (almost) i.i.d. inputs and finite lattices
For a 1D shift-invariant nearest-neighbor quantum chain, deciding the long-term average single-site state is RE-complete with a single defect, and EXPSPACE- or PSPACE-complete for finite lattices with i.i.d. inputs.