{
  "abstract": "Sampling problems give some of the clearest conditional separations between quantum and classical computation. In Fermion Sampling, passive linear optics acts on a non-Gaussian magic input. Globally Haar-random transformations give anticoncentration and high-precision average-case hardness, but require linear depth and quadratically many gates. A natural open question is whether linear depth is necessary? We show that logarithmic depth suffices for both guarantees in the same ensemble. For fresh uniform matchings with independent Haar two-mode gates, the collision reaches the passive-Haar scale at a sharp threshold of $\\log n/\\log(9/4)\\approx0.855\\log_2 n$ layers, with an explicit limiting transition profile and a matching lower bound from two-particle correlations. The mechanism is input-dependent: the magic input suppresses the extensive weight of the slowest relaxation mode, leaving the next to set the scale. An occupation-basis input retains large collision even under full Haar randomness. At a larger logarithmic depth, we prove, in the real-RAM model, average-case $\\#\\mathsf{P}$-hardness of estimating output probabilities to additive error $2^{-O(n\\log^2 n)}$ on any fixed fraction of instances above $3/4$. We construct hard instances in four native layers, embed them in typical schedules, and interpolate along Cayley paths using a rational linear-program decoder. A finite $192$-gate alphabet preserves the collision law exactly. Both guarantees use logarithmic native fermionic depth between arbitrary mode pairs and $O(n\\log n)$ gates, replacing the global-Haar construction's quadratic budget. The accuracy proved is finer than the $1/N$ scale, where $N=\\binom{n}{n/2}$, required by sampling-to-counting; hardness of sampling within constant total-variation distance remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40100",
  "author_total": 2,
  "authors": [
    {
      "name": "Natansh Mathur"
    },
    {
      "name": "Iordanis Kerenidis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aev5ygdnvhz21.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40100",
  "fulltext_url": null,
  "id": "aev5ygdnvhz21",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40100",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7215080013",
  "pdf_url": "https://arxiv.org/pdf/2609.40100",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aev5ygdnvhz21.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40100",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40100"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Logarithmic-Depth Fermion Sampling: Anticoncentration and Average-Case Hardness",
  "title_text": "Logarithmic-Depth Fermion Sampling: Anticoncentration and Average-Case Hardness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/aev5ygdnvhz21",
  "venue": "arXiv",
  "versions": []
}