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Monochromatic quantum graphs in the diagonal family

9 years old · traced to

Can a complete edge-coloured, complex-weighted graph be chosen so that perfect-matching amplitudes equal one for every monochromatic inherited vertex colouring and vanish for every nonmonochromatic colouring? The proved nonexistence family has even N=D≥4N=D\geq4.

References
  1. Open Math Problems Claimed to Be Solved with AI
quantum-information-computing
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Progress summary

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The complex-weighted diagonal case is still open: known constructions cover only a few small or two-colour cases, and no public proof or counterexample has appeared.

The question asks whether one complete edge-coloured, complex-weighted graph can produce amplitude one exactly for monochromatic inherited colourings and zero otherwise. It is part of the weighted perfect-matching problem studied by Krenn, Gu, and collaborators.

Known results

  • d=2d=2, arbitrary even nn: an alternately coloured cycle with unit edge weights works.
  • d=3d=3, n=4n=4: a unit-weight K4K_4 whose edges split into three perfect matchings works.

Current status (as of March 2026): The complex-weighted diagonal family with even N=D≥4N=D\geq 4 remains open; no verified proof of nonexistence or counterexample was found.

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