Fast counting with tensor networks

We introduce tensor network contraction algorithms for counting satisfying assignments of constraint satisfaction problems (#CSPs). We represent each arbitrary #CSP formula as a tensor network, whose full contraction yields the number of satisfying assignments of that formula, and use graph theoretical methods to determine favorable orders of contraction. We employ our heuristics for the solution of #P-hard counting boolean satisfiability (#SAT) problems, namely monotone #1-in-3SAT and #Cubic-Vertex-Cover, and find that they outperform state-of-the-art solvers by a significant margin.Comment: v2: added results for monotone #1-in-3SAT; published versio

Infinite Matrix Product States vs Infinite Projected Entangled-Pair States on the Cylinder: a comparative study

In spite of their intrinsic one-dimensional nature matrix product states have been systematically used to obtain remarkably accurate results for two-dimensional systems. Motivated by basic entropic arguments favoring projected entangled-pair states as the method of choice, we assess the relative performance of infinite matrix product states and infinite projected entangled-pair states on cylindrical geometries. By considering the Heisenberg and half-filled Hubbard models on the square lattice as our benchmark cases, we evaluate their variational energies as a function of both bond dimension as well as cylinder width. In both examples we find crossovers at moderate cylinder widths, i.e. for the largest bond dimensions considered we find an improvement on the variational energies for the Heisenberg model by using projected entangled-pair states at a width of about 11 sites, whereas for the half-filled Hubbard model this crossover occurs at about 7 sites.Comment: 11 pages, 9 figure