A polynomial-time algorithm for ground states of spin trees

TL;DR

This paper proves that ground states of local Hamiltonians on certain tree graphs obey an area law and can be efficiently computed using a novel tensor network algorithm, extending previous results beyond linear chains.

Contribution

It introduces a polynomial-time classical algorithm for finding ground states on tree graphs with fractal dimension less than 2, along with a new multi-scale tensor network structure called META-tree.

Findings

Ground states satisfy an area law on trees with fractal dimension < 2.

The algorithm efficiently computes ground states for these tree models.

Results apply to frustrated and degenerate ground states, matching line case performance.

Abstract

We prove that the ground states of a local Hamiltonian satisfy an area law and can be computed in polynomial time when the interaction graph is a tree with discrete fractal dimension $\beta<2$. This condition is met for generic trees in the plane and for established models of hyperbranched polymers in 3D. This work is the first to prove an area law and exhibit a provably polynomial-time classical algorithm for local Hamiltonian ground states beyond the case of spin chains. Our algorithm outputs the ground state encoded as a multi-scale tensor network on the META-tree, which we introduce as an analogue of Vidal's MERA. Our results hold for polynomially degenerate and frustrated ground states, matching the state of the art for local Hamiltonians on a line.