{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries",
  "title": "Depth-1 distinctness for pseudorandom unitaries",
  "canonical_statement": "A single layer of independent random single-qubit Clifford gates is $\\operatorname{negl}(n)$-distinct for polynomially many queries. Consequently, in the $PFC$ pseudorandom-unitary construction, the depth-$\\log n$ global unitary $2$-design layer can be replaced by a depth-1 tensor product of single-qubit $2$-designs while retaining the distinctness property needed for the construction. This provides a counterexample to the conjecture that every negligibly distinct ensemble must be entangling.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries:event",
      "problem_id": "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-09-02T00:00:00.000Z",
      "title": "arXiv",
      "summary": "The paper proves that a tensor-product ensemble of independently chosen single-qubit Clifford gates is $\\operatorname{negl}(n)$-distinct when the number of queries is polynomial in $n$. This disproves the authors' conjecture that negligibly distinct ensembles must necessarily be entangling. As an application, the depth-$\\log n$ global Clifford/unitary-$2$-design layer used in the $PFC$ pseudorandom-unitary construction can be replaced by a single depth-1 layer of local single-qubit $2$-designs while preserving the required distinctness property.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries:attempt"
      ],
      "method_family_ids": [
        "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries:method"
      ],
      "source_assertion_ids": [
        "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries:attempt",
      "problem_id": "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries",
      "model": "ChatGPT 5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-09-02T00:00:00.000Z",
      "human_collaborators": [
        "Asad Raza",
        "Jens Eisert",
        "Bill Fefferman"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The authors state that Proposition III.5 was proposed by ChatGPT 5.6 Sol as a counterexample to their conjecture that a $\\operatorname{negl}(n)$-distinct ensemble must be entangling. Sol identified that a single layer of independent random single-qubit Clifford gates is already negligibly distinct for polynomially many queries. ChatGPT 5.5 and 5.6 Pro were also used to devise proof strategies for the paper's main results, as well as for literature search, exposition, and technical checking; all proofs were independently verified by the authors.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2609.03065",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Depth-1 distinctness for pseudorandom unitaries",
          "url": "https://vibemathed.com/problem/depth-1-distinctness-for-pseudorandom-unitaries",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries:verification",
      "problem_id": "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries",
      "solution_event_id": "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The result appears as Proposition III.5 in a coauthored research paper by Asad Raza, Jens Eisert, and Bill Fefferman. The authors explicitly state that ChatGPT 5.6 Sol proposed the counterexample and that they independently verified all proofs. No formal proof-assistant certificate is reported.",
      "sources": [
        {
          "label": "VibeMathed: Depth-1 distinctness for pseudorandom unitaries",
          "url": "https://vibemathed.com/problem/depth-1-distinctness-for-pseudorandom-unitaries",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2609.03065",
          "kind": "primary-mathematical-source",
          "license": null
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      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2609.03065",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Depth-1 distinctness for pseudorandom unitaries",
      "url": "https://vibemathed.com/problem/depth-1-distinctness-for-pseudorandom-unitaries",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries:signal:watch_only",
      "problem_id": "vibemathed:depth-1-distinctness-for-pseudorandom-unitaries",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}