{
  "annotation_id": "7bd7a5fe-488f-4b1d-a6b8-00d282523796",
  "date_created": "2026-03-02T18:02:23.783000Z",
  "date_modified": "2026-03-02T18:02:23.783000Z",
  "file_hash": "f07cb0a316d946fe613e15e8d821cf6f5c56a115efa3c90f22b8a8405e28b01e",
  "private": false,
  "record": {
    "abstract": "The complexity of the following numerical problem is studied in the quantum\nmodel of computation: Consider a general elliptic partial differential equation\nof order 2m in a smooth, bounded domain Q\\subset \\R^d with smooth coefficients\nand homogeneous boundary conditions. We seek to approximate the solution on a\nsmooth submanifold M\\subseteq Q of dimension 0\\le d_1 \\le d. With the right\nhand side belonging to C^r(Q), and the error being measured in the L_\\infty(M)\nnorm, we prove that the n-th minimal quantum error is (up to logarithmic\nfactors) of order n^{-min((r+2m)/d_1,r/d+1)}. For comparison, in the classical\ndeterministic setting the n-th minimal error is known to be of order n^{-r/d},\nfor all d_1, while in the classical randomized setting it is (up to logarithmic\nfactors) n^{-min((r+2m)/d_1,r/d+1/2)}.",
    "arxiv_id": "quant-ph/0512241",
    "authors": [
      "Stefan Heinrich"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "The Quantum Query Complexity of Elliptic PDE",
    "url": "https://arxiv.org/abs/quant-ph/0512241"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "242aba0c-4a94-4acb-825d-a07d13355bda",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}