Breakthrough Tracker record
A claimed proof of the complex Permanent Anticoncentration Conjecture
The paper bounds the density and small-ball probability of normalized permanents in real, complex and quaternionic Ginibre ensembles. In the complex case, the authors state that this resolves the Permanent Anticoncentration Conjecture.
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- Stable ID
permanent-anticoncentration-conjecture-2026- Revision
permanent-anticoncentration-conjecture-2026.v1- Field
- Computer Science · Quantum complexity and probability
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed proof
- Last checked
AI role
The authors disclose discussions with ChatGPT, Codex assistance with writing and a Claude-suggested reference; they accept responsibility for the paper.
Record details
- Problem or result
- Aaronson–Arkhipov Permanent Anticoncentration Conjecture for complex Gaussian matrices
- Authors
- Frederic Koehler and Pui Kuen Leung
- Institutions
- University of Chicago for Koehler; Leung’s affiliation was not stated in the manuscript
- Result date
- Preprint submitted July 22, 2026
Why it matters
Permanent anticoncentration is a central assumption in arguments about the classical hardness of approximate boson sampling.
Limits
This remains a first-version preprint without located independent specialist assessment or peer review. A separate QCE 2026-accepted manuscript submitted July 24 proves only a weaker super-exponential bound and says that an inverse-polynomial Permanent Anticoncentration Conjecture remains open, but it does not cite or evaluate the Koehler–Leung manuscript posted two days earlier. The chronology and formulations require reconciliation; the parallel paper is not evidence that the claimed proof has been accepted or refuted.
Sources
- Primary: Frederic Koehler and Pui Kuen Leung, Anticoncentration of the Permanent in Ginibre Ensembles
- Parallel work: Meng, Cheng, Li and Yung, Weak Permanent Anti-Concentration and its Applications in Quantum Complexity
Correction and revision history
- 2026-07-23 — Prepared as a provisional tracker addition with the complex-case scope and downstream complexity limits stated explicitly.
- 2026-07-27 — Added a parallel QCE 2026-accepted manuscript whose weaker theorem and “open” wording do not yet reconcile with this newer claimed proof; status remains provisional.
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