{
  "abstract": "Optimistic matrix mirror-prox (OMMP) computes $ε$-approximate Nash equilibria in quantum zero-sum games with an $O(1/\\varepsilon)$ average-iterate guarantee [arXiv:2311.10859]. We investigate whether this dependence on accuracy is tight and whether geometric last-iterate convergence can be guaranteed. We study these questions through explicit games with one qubit per player. First, we prove an $Ω(1/\\varepsilon)$ lower bound for the uniform-average output that includes the maximally mixed initial state, independently of the regularizer and step size. Second, we construct a fixed game on which optimistic gradient descent-ascent (OGDA), initialized at the maximally mixed state, has last-iterate Frobenius distance to equilibrium $Θ(1/t)$ and duality gap $Θ(1/t^3)$ for every sufficiently small fixed step size. A separate fixed game exhibits arbitrarily long delays in reducing the initial error by a constant factor across a family of initial states. Finally, we give a fixed game with a unique, strictly complementary equilibrium on which optimistic matrix multiplicative weights updates (OMMWU) converge only polynomially from the maximally mixed state for every fixed positive step size. The last-iterate Frobenius distance and quantum relative entropy from the equilibrium to the iterates decay as $Θ(1/t)$, while the duality gap decays as $Θ(1/t^2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38835",
  "author_total": 3,
  "authors": [
    {
      "name": "Yiheng Su"
    },
    {
      "name": "Emmanouil-Vasileios Vlatakis-Gkaragkounis"
    },
    {
      "name": "Pucheng Xiong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8avf2yf3z388z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38835",
  "fulltext_url": null,
  "id": "8avf2yf3z388z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38835",
  "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",
      "cs.GT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7215032295",
  "pdf_url": "https://arxiv.org/pdf/2609.38835",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8avf2yf3z388z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38835",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38835"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Average-and Last-Iterate Lower Bounds for Optimistic Matrix Mirror-Prox in Quantum Zero-Sum Games",
  "title_text": "Average-and Last-Iterate Lower Bounds for Optimistic Matrix Mirror-Prox in Quantum Zero-Sum Games",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/8avf2yf3z388z",
  "venue": "arXiv",
  "versions": []
}