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We study non-equilibrium steady states (NESS) of spin chains with boundaryMarkovian dissipation from the computational complexity point of view. We focuson XX chains whose NESS are matrix product operators (MPO), i.e. withcoefficients of a tensor operator basis described by transition amplitudes inan auxiliary space. Encoding quantum algorithms in the auxiliary space, we showthat estimating expectations of operators, being local in the sense that eachacts on disjoint sets of few spins covering all the system, provides theanswers of problems at least as hard as, and believed by many computerscientists to be much harder than, those solved by quantum computers. We drawconclusions on the hardness of the above estimations.

Computational complexity of non-equilibrium steady states of quantum spin chains