Polynomial-time approximation scheme for minimum-weight decoding in color and surface code settings

Determine whether a polynomial-time approximation scheme exists for approximating the minimum-weight decoding problem in the three settings studied in the paper: (i) the triangular color code under Pauli Z noise (ColorCodeZ), (ii) the square-lattice surface code under Pauli X, Y, and Z noise (SurfaceCodeXYZ), and (iii) two square-lattice surface codes with a transversal CNOT gate under phase-flip and measurement bit-flip noise (tCNOTZ). Specifically, ascertain whether, for any fixed ε > 0, there is a polynomial-time algorithm that, given a measured syndrome, outputs an error whose Hamming weight is within a factor (1+ε) of the minimum-weight error consistent with the syndrome.

Background

The paper proves NP-hardness of minimum-weight decoding in three fundamental quantum error-correction scenarios: the color code under Z noise, the surface code under full Pauli noise, and the surface code with a transversal CNOT gate under phenomenological Z and measurement bit-flip noise. These results show that exactly computing the minimum-weight correction is intractable in these practically relevant settings.

Despite this hardness, the authors provide efficient matching-based decoders that achieve constant-factor approximations: within a factor of three for the color code and within a factor of two for the surface code and the transversal CNOT setting (Appendix A). This establishes a baseline for approximate decoding performance.

The open problem is whether one can improve these constant-factor guarantees to a polynomial-time approximation scheme (PTAS), i.e., an algorithm that, for any desired ε > 0, achieves a (1+ε)-approximation to the minimum-weight correction in polynomial time. Establishing the existence or nonexistence of such a PTAS would clarify the precise approximability of minimum-weight decoding in these canonical QEC settings.