{
  "abstract": "We consider the problem of quantum 1-PCA: given copies of an unknown $n$-qubit mixed state, recover a classical description of its leading eigenvector. Our goal is to do so using non-adaptive and single-qubit measurements. For an $n$-qubit state with top eigenvalue $λ$ and spectral gap at least $Δ> 0$, we give an algorithm that recovers the leading eigenvector to fidelity at least $1 - \\varepsilon$ with high probability using $\\tilde{O}\\left( {2^n \\cdot η^2} / {Δ^3 \\varepsilon^3}\\right)$ copies and $\\tilde{O}((2^n/Δ\\varepsilon) \\cdot \\operatorname{poly}(η/Δ\\varepsilon))$ time, where $η= \\max (1 - λ, \\varepsilon)$. All of our measurements are non-adaptively chosen, and performed in single-qubit Pauli bases. When the spectral gap is constant and the desired accuracy is comparable to the noise level, i.e. $\\varepsilon = Ω(η)$, our runtime and copy complexity become $\\tilde{O} (2^n / \\varepsilon)$. This generalizes the guarantees of Grewal et al. [arXiv:2601.04444], who achieved similar rates, but under the assumption $η= 0$, i.e., that the state was pure. Our results show that the same rates hold in the presence of state misspecification, up to polylogarithmic factors. From a technical perspective, our algorithm works by recursively constructing low-dimensional subspaces that approximately preserve the target eigenvector. To achieve nearly linear runtime dependence on the dimension of the Hilbert space, we develop a novel structured Pauli sampling scheme that enables fast batched computation of exponentially many projected Pauli matrices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06808",
  "author_total": 2,
  "authors": [
    {
      "name": "Dutch Hansen"
    },
    {
      "name": "Jerry Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/439h65n17hjdd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06808",
  "fulltext_url": null,
  "id": "439h65n17hjdd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06808",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.06808",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/439h65n17hjdd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:05Z",
      "native_id": "2610.06808",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06808"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum 1-PCA with Pauli Measurements in Nearly Linear Time",
  "title_text": "Quantum 1-PCA with Pauli Measurements in Nearly Linear Time",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:05Z",
  "url": "https://research.draconitic.com/paper/439h65n17hjdd",
  "venue": "arXiv",
  "versions": []
}