A polynomial-time approximation scheme for minimum-weight decoding of topological codes

2026-06-16

AI Summary

This work addresses the computational hardness of minimum-weight decoding for two-dimensional translation-invariant topological stabilizer codes, a key obstacle to practical fault-tolerant quantum computation. We establish, for the first time, the existence of a polynomial-time approximation scheme (PTAS) for this problem, thereby overcoming its NP-hardness barrier. By formulating decoding as a geometric optimization task that connects point-like excitations via string-like errors, we extend Arora’s framework for approximating the Euclidean traveling salesman problem to the domain of quantum error correction. Our approach applies broadly across multiple noise models and higher-dimensional topological codes. For any fixed constant ε > 0, the algorithm outputs, in polynomial time, a recovery operation whose weight is within a factor of (1 + ε) of optimal, encompassing color codes, toric codes, and quantum memories under circuit-level noise.

Abstract

Two-dimensional topological translationally invariant (2D TTI) stabilizer codes lie at the heart of fault-tolerant quantum computation, but using them requires solving the decoding problem. Minimum-weight decoding of these codes was recently shown to be NP-hard, even in basic settings, such as the color code with Pauli $Z$ errors and the toric code with Pauli $X$, $Y$ and $Z$ errors. Here, we prove that minimum-weight decoding of 2D TTI codes nonetheless admits a polynomial-time approximation scheme (PTAS), i.e., for any constant $\varepsilon>0$, a recovery operator of weight within a multiplicative factor of $1+\varepsilon$ of the minimum can be found in polynomial time. Our approach builds on Arora's PTAS for Euclidean problems, such as the traveling salesman problem, and applies when decoding can be cast in terms of point-like excitations connected by string-like errors. It therefore extends beyond two dimensions, covering certain higher-dimensional topological codes and quantum memories, including the toric code with phenomenological or circuit-level noise.

Problem

• topological codes
• minimum-weight decoding
• NP-hard
• 2D TTI stabilizer codes
• fault-tolerant quantum computation

Innovation

• polynomial-time approximation scheme
• topological codes
• minimum-weight decoding
• fault-tolerant quantum computation
• NP-hard