Hardness of approximation for minimum-weight decoding of two-dimensional topological quantum codes

Abstract:Efficient decoding is essential for the practical realization of fault-tolerant quantum computers. We study the computational complexity of minimum-weight decoding for topological quantum codes. For surface codes under the depolarizing channel, we consider Minimum-Weight decoding, which seeks a minimum-weight Pauli error consistent with both the $X$- and $Z$-syndromes. For color codes under independent $X$- and $Z$-error models, we consider Separate Minimum-Weight decoding. Assuming $P\neq NP$, we establish polynomial additive inapproximability gaps for these problems. Specifically, for the toric code and the $4.8.8$ color code on the torus, no polynomial-time algorithm can always produce a solution whose weight is within $\Omega(N^{1/14})$ of the optimum, where $N$ is the number of qubits. For the planar surface code, we obtain an $\Omega(N^{1/18})$ gap. Our inapproximability results use Håstad's hardness of approximation for MAX-3SAT. Our reduction develops a general, modular framework for embedding logical constraints into coupled primal--dual join problems on a lattice. A key ingredient is a localization argument that controls unintended interactions between different parts of the construction.

Quantum Physics (quant-ph); Computational Complexity (cs.CC)

arXiv:2608.17109 [quant-ph]

(or arXiv:2608.17109v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.17109 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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