{"paper":{"title":"CNOT-Distance is NP-complete under all-to-all connectivity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"quant-ph","authors_text":"Antonio Acuaviva, Arturo Acuaviva, Pablo Acuaviva","submitted_at":"2026-08-04T15:34:26Z","abstract_excerpt":"Given $A\\in\\operatorname{GL}(N,2)$ and an integer $K$, we ask whether $A$ can be implemented by at most $K$ CNOT gates on fixed labelled wires with all-to-all connectivity. We prove that this problem is NP-complete. From a finite simple graph $G=(V,E)$, we construct an upper-unitriangular matrix $A_G\\in\\operatorname{GL}(2|V|+|E|+1,2)$ satisfying $\\ell_{\\mathrm{CNOT}}(A_G)=2|V|+2|E|+\\tau(G)$, where $\\tau(G)$ is the minimum vertex-cover size. Each target matrix has $O(N)$ nonzero entries and row Hamming weight at most four. The lower bound unfolds an arbitrary CNOT circuit into an XOR directed a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.03825","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.03825/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}