This paper shows that for a large class of two-dimensional topological quantum error‑correcting codes, one can find a recovery (an error correction) whose size is within any small factor 1+ε of the true minimum, and do so in polynomial time. The authors prove a polynomial‑time approximation scheme (PTAS) for the minimum‑weight decoding problem. That problem asks: given the pattern of measured parity checks (the syndrome), find a smallest set of physical errors that could have caused it.

The result applies to two‑dimensional translationally invariant stabilizer codes on an L×L lattice. Examples of such codes include the toric code and related 2D codes. The algorithm guarantees that for any fixed ε>0 it returns a recovery of weight at most (1+ε) times the minimum. The randomized version runs in time roughly L^2 (log L)^{O(1/ε)} and succeeds with probability at least 1/4. A deterministic (derandomized) version always succeeds and runs in time about L^4 (log L)^{O(1/ε)}. The paper also explains how the approach extends to certain higher‑dimensional codes and to models of noise such as phenomenological or circuit‑level noise for the toric code.

The construction builds on Arora’s PTAS ideas for geometric problems like the traveling salesman problem. The decoding problem is viewed in terms of point‑like excitations (often called anyons) that are connected or moved by string‑like error operators. The lattice is recursively cut into squares. The authors argue that an optimal error can be slightly reshaped so it crosses square boundaries only at a fixed set of regularly spaced “portals” and only a bounded number of times. That restricted form can then be found by dynamic programming on the dissection tree, giving the near‑optimal solution efficiently. The proof uses two key operations called rerouting and patching to transform an optimal solution into the portal‑respecting form with only a small weight increase.