Research Preprints · Complex geometry

Lower bounds for the quantum permanent

Suvrit Sra · TU Munich, Department of Mathematics, School of CIT

Version v1 · released 24 October 2026 · 30 pages

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Abstract

Griffiths's 1969 conjecture on the positivity of Schur forms reduces to the positivity of the quantum permanent $\Phi_n$ on positive bistochastic maps. This conjecture was refuted by Du with a map for which $\Phi_9<0$. Therefore, we ask: how negative can $\Phi_n$ be? We prove a lower bound $\Phi_n\ge-3n/(3n^2+20n+103)$ (for $n\ge 8$), and an asymptotically stronger bound $\Phi_n\ge-1/n+(3e^2-7-o(1))\log n/n^2$. The key to our bounds is the orbital polynomial of the map, which turns $\Phi_n$ into a linear function of a distribution on partitions and positivity into linear constraints. We also construct explicit maps for $n=8, 9$ that improve on Du's example. Finally, if $d_*\in\{4,\ldots,8\}$ is the first dimension for which $\Phi_n<0$, the minimum $\mu_n$ of $\Phi_n$ over positive bistochastic maps satisfies $-\mu_n=\Omega(n^{-d_*})$; we conjecture the matching order $-\mu_n=O(n^{-d_*})$.

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@misc{sra2026qperm-lower-v1,
  author       = {Suvrit Sra},
  title        = {{Lower bounds for the quantum permanent}},
  year         = {2026},
  month        = oct,
  howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/qperm-lower/v1}},
  note         = {Version v1, released 24 October 2026},
}

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