{
  "abstract": "`Solvable' circuits, such as dual unitaries and its generalisations, have arisen as paradigmatic examples of tractable chaotic non-equilibrium dynamics, both in classical and quantum systems. However, while increasingly complicated sufficient conditions have been proposed, a systematic theory classifying and understanding general features of Solvable circuits is missing. In this work we develop such a theory by deriving the most general local necessary and sufficient conditions under which a $(1+1)D$ circuit is Solvable. Concretely, we define a circuit to be \\emph{influence-solvable} if its influence matrix, representing the `bath' generated by its own evolution, is given by a uniform MPS with finite bond-dimension $χ$. This property allows for efficient computation of subsystem dynamics and essentially contains all known examples of Solvable circuits. The set of necessary and sufficient local conditions follows from a new version of the fundamental theorem of MPS for open boundary conditions. Next, we apply our theory to brickwork circuits with $χ=1$ influence-solvability and perform a systematic classification of classical brickwork circuits with local dimension up to $d=3$ and quantum brickwork circuits with $d=2$. Our search reveals new Solvable circuits that are not captured by known conditions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.12538",
  "author_total": 2,
  "authors": [
    {
      "name": "Friedrich Hübner"
    },
    {
      "name": "Sun Woo P. Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4gq0mnd61f4s5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.12538",
  "fulltext_url": null,
  "id": "4gq0mnd61f4s5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.12538",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Quantum many-body systems"
    ]
  ],
  "openalex": "W7164478653",
  "pdf_url": "https://arxiv.org/pdf/2606.12538",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4gq0mnd61f4s5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2606.12538",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.12538"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Influence-solvability: a systematic theory of $(1+1)D$ solvability and its application to brickwork circuits",
  "title_text": "Influence-solvability: a systematic theory of (1+1)D solvability and its application to brickwork circuits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:06Z",
  "url": "https://research.draconitic.com/paper/4gq0mnd61f4s5",
  "venue": "arXiv",
  "versions": []
}