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We present a unified exact tensor network approach to compute the groundstate energy, identify the optimal configuration, and count the number ofsolutions for spin glasses. The method is based on tensor networks with theTropical Algebra defined on the semiring. Contracting the tropical tensornetwork gives the ground state energy; differentiating through the tensornetwork contraction gives the ground state configuration; mixing the tropicalalgebra and the ordinary algebra counts the ground state degeneracy. Theapproach brings together the concepts from graphical models, tensor networks,differentiable programming, and quantum circuit simulation, and easily utilizesthe computational power of graphical processing units (GPUs). For applications,we compute the exact ground state energy of Ising spin glasses on squarelattice up to 1024 spins, on cubic lattice up to 216 spins, and on 3 regularrandom graphs up to 220 spins, on a single GPU; We obtain exact ground stateenergy of (+/-)J Ising spin glass on the chimera graph of D-Wave quantumannealer of 512 qubits in less than 100 seconds and investigate the exact valueof the residual entropy of (+/-)J spin glasses on the chimera graph; Finally,we investigate ground-state energy and entropy of 3-state Potts glasses onsquare lattices up to size 18 x 18. Our approach provides baselines andbenchmarks for exact algorithms for spin glasses and combinatorial optimizationproblems, and for evaluating heuristic algorithms and mean-field theories.

Tropical Tensor Network for Ground States of Spin Glasses