Tropical Tensor Network for Ground States of Spin Glasses

We present a unified exact tensor network approach to compute the ground state energy, identify the optimal configuration, and count the number of solutions for spin glasses. The method is based on tensor networks with the tropical algebra defined on the semiring of $(\mathbb{R}\cup \{-\infty \},\oplus ,\odot )$. Contracting the tropical tensor network gives the ground state energy; differentiating through the tensor network contraction gives the ground state configuration; mixing the tropical algebra and the ordinary algebra counts the ground state degeneracy. The approach brings together the concepts from graphical models, tensor networks, differentiable programming, and quantum circuit simulation, and easily utilizes the computational power of graphical processing units (GPUs). For applications, we compute the exact ground state energy of Ising spin glasses on square lattice up to 1024 spins, on cubic lattice up to 216 spins, and on three regular random graphs up to 220 spins, on a single GPU; we obtain exact ground state energy of $\pm J$ Ising spin glass on the chimera graph of D-Wave quantum annealer of 512 qubits in less than 100 s and investigate the exact value of the residual entropy of $\pm J$ spin glasses on the chimera graph; finally, we investigate ground-state energy and entropy of three-state Potts glasses on square lattices up to size $18\times 18$. Our approach provides baselines and benchmarks for exact algorithms for spin glasses and combinatorial optimization problems, and for evaluating heuristic algorithms and mean-field theories.

Figure 1

(a) The tensor network representation of a square lattice Ising spin glass. (b) An equivalent circuit representation used for the practical simulation. See text for the definition of the symbols.

Figure 2

(a) A chimera lattice with $4\times 4$ unit cells. Dots represent Ising spins and lines indicate couplings. (b) Tensor network representation, where each node has a degree of freedom of four spins. (c) Wall clock time for computing the ground state energy of Ising spin glass on the chimera graph with the $L\times L$ unit cell ($8L^2$ spins).

Figure 3

(a) Histogram of the ground state degeneracy of $\pm J$ spin glasses on the chimera graph with $L\times L$ unit cells ($8L^2$ Ising spins). For each system size, we solve 10000 random instances. (b) The residual entropy density versus system size.