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We develop a technique for calculating three-dimensional classical partitionfunctions using projected entangled-pair states (PEPS). Our method is based onvariational PEPS optimization algorithms for two-dimensional quantum spinsystems, and allows us to compute free energies directly in the thermodynamiclimit. The main focus of this work is classical frustration inthree-dimensional many-body systems leading to an extensive ground-statedegeneracy. We provide high-accuracy results for the residual entropy of thedimer model on the cubic lattice, water-ice $I_h$ and water-ice $I_c$. Inaddition, we show that these systems are in a Coulomb phase by computing thedipolar form of the correlation functions. As a further benchmark of ourmethods, we calculate the critical temperature and exponents of the Ising modelon the cubic lattice.

Residual entropies for three-dimensional frustrated spin systems with tensor networks