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Projected entangled pair states (PEPS) offer memory-efficient representationsof some quantum many-body states that obey an entanglement area law, and arethe basis for classical simulations of ground states in two-dimensional (2d)condensed matter systems. However, rigorous results show that exactly computingobservables from a 2d PEPS state is generically a computationally hard problem.Yet approximation schemes for computing properties of 2d PEPS are regularlyused, and empirically seen to succeed, for a large subclass of (not tooentangled) condensed matter ground states. Adopting the philosophy of randommatrix theory, in this work we analyze the complexity of approximatelycontracting a 2d random PEPS by exploiting an analytic mapping to an effectivereplicated statistical mechanics model that permits a controlled analysis atlarge bond dimension. Through this statistical-mechanics lens, we argue that:i) although approximately sampling wave-function amplitudes of random PEPSfaces a computational-complexity phase transition above a critical bonddimension, ii) one can generically efficiently estimate the norm andcorrelation functions for any finite bond dimension. These results aresupported numerically for various bond-dimension regimes. It is an importantopen question whether the above results for random PEPS apply more generallyalso to PEPS representing physically relevant ground states

Random insights into the complexity of two-dimensional tensor network calculations