SpinGlassPEPS.jl: Tensor-network package for Ising-like optimization on quasi-two-dimensional graphs

TL;DR

SpinGlassPEPS.jl targets Ising-like and QUBO optimization on quasi-2D graphs by employing PEPS to approximate the Boltzmann distribution and drive a branch-and-bound search over probable configurations. The package is modular, comprising SpinGlassEngine.jl, SpinGlassNetworks.jl, and SpinGlassTensors.jl, with configurable contraction schemes and GPU acceleration to handle large, sparse tensor networks. It maps Ising/QUBO instances to Potts Hamiltonians on king's graphs, computes marginal probabilities via approximate PEPS contractions, and reconstructs the low-energy spectrum including ground states and excitations. Benchmark results on large king's-graph problems indicate competitive performance against CPLEX and, in some cases, superiority to SBM, highlighting the practical relevance for quantum and classical annealing architectures.

Abstract

This work introduces SpinGlassPEPS$.$jl, a software package implemented in Julia, designed to find low-energy configurations of generalized Potts models, including Ising and QUBO problems, utilizing heuristic tensor network contraction algorithms on quasi-2D geometries. In particular, the package employs the Projected Entangled-Pairs States to approximate the Boltzmann distribution corresponding to the model's cost function. This enables an efficient branch-and-bound search (within the probability space) that exploits the locality of the underlying problem's topology. As a result, our software enables the discovery of low-energy configurations for problems on quasi-2D graphs, particularly those relevant to modern quantum annealing devices. The modular architecture of SpinGlassPEPS$.$jl supports various contraction schemes and hardware acceleration.

Figures (5)

• Figure 1: Execution flow. The Ising problem in (a) is mapped to a Potts Hamiltonian defined on a king's graph in (b). This allows the partition function of that Hamiltonian to be represented as a PEPS tensor network on a square lattice, as in (c). The main algorithm executes the branch and bound search in the probability space, building the most probable configurations by adding one Potts variable at a time. The marginal conditional probabilities follow from an approximate contraction of the corresponding tensor network in (d). The full branch and bound sweep results in a candidate for the most probable (ground state) configuration in (e), together with a set of localized excitations on top of it in (f).
• Figure 2: Interoperability between all SpinGlassPEPS.jl packages. The input Ising problem file is processed by SpinGlassNetworks.jl, transforming it into a Potts Hamiltonian. The latter is then passed to SpinGlassEngine.jl, together with solver's and contraction's parameters. The SpinGlassEngine.jl module serves as the core branch and bound solver. It passes the problem of marginal probabilities' calculation to SpinGlassTensors.jl, that constructs and approximately contracts the corresponding tensor network. Finally, the solution (ground state, excitations, and their energies) is returned to the user as an output.
• Figure 3: Benchmarking SpinGlassPEPS.jl for two sets of Ising problems defined on graphs from Fig. \ref{['fig:1']}(a) with $N = 50 \times 50 \times d$ spins, for $d = 1$ (top row) and $d=2$ (bottom row). As reference solvers, We employ a Simulated Bifurcation machine (SBM) Goto2019 and CPLEX. Panels (a) and (d) show time to solution to the best energy for a median instance ($E_{\rm best}$ is the best results among the considered solvers), and in (b) and (e), we show instance-wise results for $10$ instances. In (c) and (f), w2e show the diversity of obtained solutions, i.e., the number of solutions within approximation ratio $a_r = 0.01$ ($E - E_{\rm best} < a_r \cdot 2 \cdot E_{\rm best}$), where each pair has a Hamming distance greater than $N / 2$.
• Figure 4: Problems defined on Chimera, Pegasus, and Zephyr graphs Boothby2021Boothby2020Dattani2019Lanting2014, employed in the past and current D-Wave quantum annealers, can be mapped to the Potts Hamiltonian on king's graph upon grouping $8$, $24$, and $16$ spins, respectively. Due to the large unit cell size of the latter two graphs, they require further processing. In particular, sparse connectivity structures between unit cells and GPU acceleration. Both are supported by SpinGlassPEPS.jl package.
• Figure 5: The picture in panel (a) shows the used inpainting problem. Data is given on the circular boundary, and the solution should fill in the black region. It is part of the OpenGM2 Benchmark dataset openGM. The picture in panel (b) shows the ground state obtained by SpinGlassPEPS.jl. Due to the introduced anisotropy by the grid used in discretization, the results show a bias toward axis-parallel edges Lellmann2011. Yellow region in (c) shows a low-energy excitation, i.e., a group of variables that should be collectively filliped to obtain another low-energy solution.