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    "story_id": "mp-2026-09-06-014",
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    "headline": "Even the Plain CNOT Circuit Problem Is NP-Hard",
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    "dek": "A reduction from Hamiltonian paths closes the complexity question for exact synthesis with labeled qubits, all-to-all links and no ancillas.",
    "summary": "A reduction from Hamiltonian paths closes the complexity question for exact synthesis with labeled qubits, all-to-all links and no ancillas.",
    "body_text": "Earlier hardness proofs needed restricted connectivity, encoded inputs or extra intermediate variables. The new proof uses recorder qubits to force required intermediate visits into the final parity transformation, reducing a grid-graph Hamiltonian path to the vanilla synthesis problem. The decision form is NP-complete and optimization is NP-hard, with consequences for related shortest-word, Cayley-graph distance and XOR-program problems. Complexity hardness describes worst-case computation; it does not say useful circuits cannot be optimized in practice.",
    "why_it_matters": "A reduction from Hamiltonian paths closes the complexity question for exact synthesis with labeled qubits, all-to-all links and no ancillas.",
    "limitations": [
      "Complexity hardness describes worst-case computation; it does not say useful circuits cannot be optimized in practice."
    ],
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        "text": "A reduction from Hamiltonian paths closes the complexity question for exact synthesis with labeled qubits, all-to-all links and no ancillas.",
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        "qualification": "Complexity hardness describes worst-case computation; it does not say useful circuits cannot be optimized in practice."
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    "tags": [
      "quantum circuits",
      "complexity",
      "CNOT synthesis"
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  "sources": [
    {
      "source_id": "source-2026-09-06-014",
      "title": "arXiv preprint 2609.04160",
      "publisher": "arXiv",
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      "published_at": "2026-09-03T13:49:16.000-04:00",
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    "title": "Even the Plain CNOT Circuit Problem Is NP-Hard",
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