{
  "abstract": "We give a quantum algorithm for stochastic $2$-player games. Their game trees are the expectiminimax trees of classical game search: $m$ adversarial layers alternate with $m$ chance layers, every node has $deg$ children, and the leaf values lie in an interval of length $N$. The algorithm estimates the value of the game to root mean square error $ε$ with \\[ O\\!\\left( K^{D}\\, deg^{\\frac{D}{4}} \\, \\frac{N}ε \\, D^{4D} \\log\\left( \\frac{deg N}ε \\right)^{4D} \\right), \\qquad D = 2m \\] queries to the leaf values, where $K$ is an absolute constant. If we treat the depth $D$ as a constant as is sometimes done in the literature, the bound is $\\widetilde O(deg^{m/2}ε^{-1})$ against the classical $\\widetilde O(deg^{m}ε^{-2})$. So the quadratic speedup in the branching factor known for adversarial trees and the quadratic speedup in the accuracy known for a single expectation both survive when we nest one inside the other. We use a derandomised multilevel Monte Carlo estimator for chance layers and a coherent binary search for adversarial layers. The composition is done by introducing a conversion between root-mean-square and uniform error guarantees, and a composition to nest it with quantum mean estimation. We state the algorithm for stochastic games, but it is valid for the following conditions: the value of a parent is a Lipschitz function of the values of its children and the value at a chance vertex is linear in its children, however the speedup is preserved only if the deterministic step has a quantum subroutine with sublinear query complexity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35511",
  "author_total": 5,
  "authors": [
    {
      "name": "Marien Raat"
    },
    {
      "name": "Merlin Incerti-Medici"
    },
    {
      "name": "James R. Wootton"
    },
    {
      "name": "Evert van Nieuwenburg"
    },
    {
      "name": "Daniel Bultrini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dr7c9m5ksdptf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35511",
  "fulltext_url": null,
  "id": "dr7c9m5ksdptf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35511",
  "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": "W7214773807",
  "pdf_url": "https://arxiv.org/pdf/2609.35511",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dr7c9m5ksdptf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2609.35511",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35511"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "How fast can a parent estimate the value of their children? A quantum algorithm for stochastic games",
  "title_text": "How fast can a parent estimate the value of their children? A quantum algorithm for stochastic games",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:12Z",
  "url": "https://research.draconitic.com/paper/dr7c9m5ksdptf",
  "venue": "arXiv",
  "versions": []
}