Quantum complexity and generalized area law in fully connected models

Abstract: The area law for entanglement entropy fundamentally reflects the complexity of quantum many-body systems, demonstrating ground states of local Hamiltonians to be represented with low computational complexity. While this principle is well-established in one-dimensional systems, little is known beyond 1D cases, and attempts to generalize the area law on infinite-dimensional graphs have largely been disproven. In this work, for non-critical ground states of Hamiltonians on fully connected graphs, we establish a generalized area law up to a polylogarithmic factor in system size, by effectively reducing the boundary area to a constant scale for interactions between subsystems. This result implies an efficient approximation of the ground state by the matrix product state up to an approximation error of $1/\text{poly}(n)$. As the core technique, we develop the mean-field renormalization group approach, which rigorously guarantees efficiency by systematically grouping regions of the system and iteratively approximating each as a product state. This approach provides a rigorous pathway to efficiently simulate ground states of complex systems, advancing our understanding of infinite-dimensional quantum many-body systems and their entanglement structures.