Simulation of three-dimensional quantum systems with projected entangled-pair states

Abstract

Tensor network algorithms have proven to be very powerful tools for studying one-and twodimensional quantum many-body systems. However, their application to three-dimensional (3D) quantum systems has so far been limited, mostly because the efficient contraction of a 3D tensor network is very challenging. In this paper we develop and benchmark two contraction approaches for infinite projected entangled-pair states (iPEPS) in 3D. The first approach is based on a contraction of a finite cluster of tensors including an effective environment to approximate the full 3D network. The second approach performs a full contraction of the network by first iteratively contracting layers of the network with a boundary iPEPS, followed by a contraction of the resulting quasi-2D network using the corner transfer matrix renormalization group. Benchmark data for the Heisenberg and Bose-Hubbard models on the cubic lattice show that the algorithms provide competitive results compared to other approaches, making iPEPS a promising tool to study challenging open problems in 3D.

Tensor networks and efficient descriptions of classical data

“…In this case, area law scaling would indicate that the two-dimensional projected entangled pair states (PEPS) TNs [55] with a bond dimension mildly growing with system size would be an adequate representation. For quantum many-body systems, these TNs have been shown to be effective at representing two-dimensional gaped local systems, which also exhibit area law behavior [56][57][58][59][60][61][62][63]. Earlier works have studied MI scaling in the MNIST data set, with somewhat contradictory results [64][65][66].…”