Thomas Barthel

Cost scaling for matrix-product and tree-tensor-network simulations of 2D and 3D systems with area-law entanglement

Tensor network states are an indispensable tool for the simulation of strongly correlated quantum many-body systems. In recent years, tree tensor network states (TTNS) have been successfully used for the investigation of two-dimensional systems, also to benchmark quantum simulation approaches for condensed matter, nuclear, and particle physics. In comparison to the more traditional approach based on matrix product states (MPS), the graph distance of physical degrees of freedom can be drastically reduced in TTNS. In extension of corresponding results for MPS [Verstraete and Cirac, Phys. Rev. B 73, 094423 (2006)], I will show how to bound TTNS approximation errors using Schmidt spectra or Renyi entanglement entropies of the target quantum state. Conversely, one obtains bounds on tensor-network bond dimensions needed to achieve a specific approximation accuracy. Surprisingly, it turns out that, asymptotically, MPS simulations of low-energy states should be more efficient than TTNS simulations for both two-dimensional and three-dimensional systems. I will discuss the computational complexity for different boundary conditions under the assumption that the system obeys an entanglement area law in the sense that bond dimensions scale exponentially in the surface area of the associated subsystem.