Cost Scaling for Matrix-Product and Tree-Tensor-Network Simulations of 2D and 3D Systems with Area-Law Entanglement

Explore the computational complexity and cost scaling of tensor network simulations in this 50-minute conference talk examining matrix-product states (MPS) and tree-tensor-network states (TTNS) for 2D and 3D quantum systems. Learn how tensor network states serve as essential tools for simulating strongly correlated quantum many-body systems, with particular focus on how TTNS have been successfully applied to investigate two-dimensional systems and benchmark quantum simulation approaches across condensed matter, nuclear, and particle physics. Discover how TTNS can drastically reduce the graph distance of physical degrees of freedom compared to traditional MPS approaches. Examine the mathematical framework for bounding TTNS approximation errors using Schmidt spectra or Renyi entanglement entropies of target quantum states, and understand how these bounds translate to tensor-network bond dimension requirements for achieving specific approximation accuracy. Investigate the surprising finding that MPS simulations of low-energy states may be asymptotically more efficient than TTNS simulations for both two-dimensional and three-dimensional systems. Analyze computational complexity considerations for different boundary conditions under the assumption that systems obey an entanglement area law, where bond dimensions scale exponentially with the surface area of associated subsystems.