{
  "annotation_id": "d5c038ff-f638-4c09-86e6-d75edc4b3551",
  "date_created": "2026-03-02T18:02:44.190000Z",
  "date_modified": "2026-03-02T18:02:44.190000Z",
  "file_hash": "34374b05c733d43be9597a03bdfe274e3afd269fffe551f0f43e248ee3fc0c66",
  "private": false,
  "record": {
    "abstract": "We isolate and generalize a technique implicit in many quantum algorithms,\nincluding Shor\u0027s algorithms for factoring and discrete log. In particular, we\nshow that the distribution sampled after a Fourier transform over ${\\mathbb\nZ}_p$ can be efficiently approximated by transforming over ${\\mathbb Z}_q$ for\nany q in a large range. Our result places no restrictions on the superposition\nto be transformed, generalizing the result implicit in Shor which applies only\nto periodic superpositions. In addition, our proof easily generalizes to\nmulti-dimensional transforms for any constant number of dimensions.",
    "arxiv_id": "quant-ph/9812060",
    "authors": [
      "Lisa Hales",
      "Sean Hallgren"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "Sampling Fourier Transforms on Different Domains",
    "url": "https://arxiv.org/abs/quant-ph/9812060"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "e220deb8-457d-4647-b59c-46c539d62ac6",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}