Efficient tensor-network approximation in 2D

This is an existence question about a pure-state representation, not an algorithm for discovering it or contracting it. The polynomial may depend on the fixed local dimension, range, interaction bound and gap lower bound, but not on system size. Global vector-norm error is deliberately stronger than reproducing bounded-region reduced states.

What is known: In 1D, an area law plus a spectral gap yields an efficient MPS approximation, and the ground state can be found in polynomial time (Landau–Vazirani–Vidick). In 2D the chain of implications breaks: bounded entropy across cuts does not obviously bound the bond dimension needed for a global PEPS, and contracting a general 2D PEPS is itself #P-hard (see B3), so even having a PEPS description need not make it useful.

A distinct weaker target asks for trace-distance error at most $\epsilon$ on every region of a fixed size. Its scaling must be stated separately; allowing mixed or purified tensor networks can also change the task. Gibbs-state PEPS approximations and local-observable approximation theorems do not automatically establish the global pure-state statement. An area-law bound on von Neumann entropy alone is not a universal efficient-representation theorem.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.