Metric Self-Dual Completion and Optimal Additive Hardness for Quantum and Graph-State Distance

@misc{ostrovsky2026metric,
  title = {{Metric Self-Dual Completion and Optimal Additive Hardness for Quantum and Graph-State Distance}},
  author = {Rafail Ostrovsky},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22669},
  eprint = {2609.22669},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22669},
}

TY  - UNPB
TI  - Metric Self-Dual Completion and Optimal Additive Hardness for Quantum and Graph-State Distance
AU  - Rafail Ostrovsky
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.22669
UR  - https://arxiv.org/abs/2609.22669
AB  - We prove that the quantum code distance is NP-hard to approximate within an additive error of $c N$, for some constant c >0, where N is the number of qubits. Our reductions are deterministic. This improves the previous square-root additive gap to Ω(N) and resolves the explicitly stated linear-gap question of Kapshikar and Kundu. Our result holds for CSS codes with identical X- and Z-check spaces, and with a constant rate and constant relative distance. For every fixed λ>1, there is a constant c>0 such that hardness still holds even when every nonidentity stabilizer has weight greater than λ times the quantum distance. We also improve the hardness gap of graph state distance on N vertices of Grigorescu, Jha, and Samperton from cube-root to Ω(N), resolving their explicitly stated open question. Both hardness results are asymptotically optimal since both distances are at most N. Our graph state distance hardness result holds for balanced bipartite graphs with a binary adjacency matrix that is its own inverse (mod 2). Our main technique for both hardness bounds above is classical: we show how to convert any code C of length m into a self-dual code A(C) of length N=Θ(m) while exactly doubling the original coset metric. The conversion is deterministic and efficient. We call it the metric self-dual completion of C. It comes with a linear embedding τ: 𝔽_2^m ↪ 𝔽_2^N. The embedding doubles all Hamming distances between vectors in 𝔽_2^m and all pairwise distances between corresponding cosets. The embedding also guarantees that all codewords of A(C) of weight at most 2m are exactly τ(C).
ER  -