{
  "abstract": "We present a family of quantum amplitude amplification algorithms that interleave ordinary Grover rotations with tunable weak measurements in loops. A successful measurement yields the target state and stops the algorithm, while after a failure, the loop resumes from the post-measurement state without restarting. We express the leading costs in expected Grover iterations as $p \\to 0$. For a known target probability $p$, an exact weak measurement-conditioned loop for states near the angle $π/4$ uses $(π/8+1/4+o(1))/\\sqrt{p}$ iterations, improving on the standard Grover constant $π/4$ and on optimised restart Grover, and matching the corresponding infimum in our continuous-angle analysis. When only a lower bound $0 < p_0 \\leq p < 1/2$ is available, a discrete Lyapunov equation gives the exact expected oracle cost for every fixed measurement strength and a closed-form optimal strength. Using the strength selected from $p_0$, the expected number of Grover iterations is at most $\\frac{1}{2\\sqrt{2}}(\\frac{1}{\\sqrt{p}}+\\frac{1}{\\sqrt{p_0}})$, where the expected cost decreases as the actual $p$ increases, and gives the leading constant $1/\\sqrt{2}$ at the promise boundary $p = p_0$. For completely unknown $p$, we identify measurement strength $κ(t)=Θ(1/t)$ as the critical scale within the regular schedules considered here and analyse $κ_b(t)=\\min\\{1/2,b/t\\}$. For each fixed $b > 2$, we rigorously derive a closed-form Gamma-function expression $C(b)$ giving $(C(b)/2+o(1))/\\sqrt{p}$ expected iterations. Numerical minimisation of this explicit expression gives $b \\approx 5.2$ and $C(b)/2 \\approx 1.01$. The results show that weak measurements preserve the $Θ(1/\\sqrt{p})$ search scale while adding an explicit fixed-point control mechanism and provable oracle savings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39621",
  "author_total": 2,
  "authors": [
    {
      "name": "Wang Fang"
    },
    {
      "name": "Chris Heunen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/428ceb3khz2x8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39621",
  "fulltext_url": null,
  "id": "428ceb3khz2x8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39621",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7215034083",
  "pdf_url": "https://arxiv.org/pdf/2609.39621",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/428ceb3khz2x8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39621",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39621"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Weakly Measured Loops for Quantum Amplitude Amplification: Oracle Savings and Adaptive Search with Unknown Target Probability",
  "title_text": "Weakly Measured Loops for Quantum Amplitude Amplification: Oracle Savings and Adaptive Search with Unknown Target Probability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/428ceb3khz2x8",
  "venue": "arXiv",
  "versions": []
}