{
  "abstract": "Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) $\\frac{\\mathrm{d}}{\\mathrm{d} t} ρ_t = \\mathcal{L} ρ_t$, where $\\mathcal{L}$ is a linear (super-)operator and $ρ_t$ is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state $ρ_\\infty$ (such that $\\mathcal{L} ρ_\\infty = 0$), the first few excited states, and trajectories $t\\mapsto ρ_t$. In finite dimension, $ρ_t$ is an $n\\times n$ matrix, $\\mathcal{L}$ thus typically costs $n^4$ to store explicitly as a dense matrix, and $O(n^6)$ to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, $\\mathcal{L}$ usually costs only $O(n^3)$ to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution $\\mathcal{S}$, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map $Φ$ whose fixed point is directly related to $ρ_\\infty$, all the other eigenvalues having smaller magnitude. The map $Φ$ is thus well suited to iterative methods, and $ρ_\\infty$ can be found in a few Arnoldi iterations. Using the same inverse map $\\mathcal{S}^{-1}$ as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like $O(n^3)$ per iteration and offer state-of-the-art performance on CPU and GPU.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30860",
  "author_total": 4,
  "authors": [
    {
      "name": "Gaspard Beugnot"
    },
    {
      "name": "Paul Gregory"
    },
    {
      "name": "Rémi Robin"
    },
    {
      "name": "Antoine Tilloy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fq8d6g33qycr3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30860",
  "fulltext_url": null,
  "id": "fq8d6g33qycr3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30860",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7204932694",
  "pdf_url": "https://arxiv.org/pdf/2608.30860",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fq8d6g33qycr3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "2608.30860",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30860"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Well-conditioned iterative methods for large open quantum systems",
  "title_text": "Well-conditioned iterative methods for large open quantum systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/fq8d6g33qycr3",
  "venue": "arXiv",
  "versions": []
}