Tropical Tensor Network for Ground States of Spin Glasses

TL;DR

This paper introduces a novel tensor network method using Tropical Algebra to exactly compute ground states and degeneracies of spin glasses, leveraging GPU acceleration for large and complex systems.

Contribution

The paper develops a unified tropical tensor network approach that combines graphical models, tensor networks, and differentiable programming for spin glass analysis.

Findings

Computed ground state energies for large spin glasses on various lattices.

Achieved fast exact solutions for D-Wave chimera graphs with 512 qubits.

Provided benchmarks for spin glass and optimization algorithms.

Abstract

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. 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 3 regular random graphs up to 220 spins, on a single GPU; We obtain exact ground state energy of (+/-)J Ising spin glass on the chimera graph of D-Wave quantum annealer of 512 qubits in less than 100 seconds and investigate the exact value of the residual entropy of (+/-)J spin glasses on the chimera graph; Finally, we investigate ground-state energy and entropy of 3-state Potts glasses on square lattices up to size 18 x 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.