Transverse spectrum governing the finite-depth ceiling

Characterize the transverse spectrum of the peakedness objective on the gauge quotient, in order to determine the finite-depth ceiling $\delta^*$ beyond the exactly flat gauge directions.

Background

The paper derives a substantial family of exact flat directions caused by redundancies in the trainable circuit parametrization and by boundary transformations that leave the probe ray, and hence the peakedness objective, unchanged. Hessian measurements at several sizes confirm the predicted flat-band dimension.

The authors state that the flat-band analysis does not determine the global finite-depth maximum. They identify the remaining transverse Hessian spectrum on the quotient by the gauge directions as the unresolved object needed to understand δ∗\delta^*.

References

The local geometry at the solutions is closed here, the flat band being the objective's gauge in closed form at every size, which moves the open question about $\delta*$ to the transverse spectrum on the gauge quotient, of dimension $P - 3S - 9B_2 - 4B_1$, computed at three sizes and modeled at none.

— The optimization landscape of peaked-circuit generation  (2608.11890 - Jamoussi, 12 Aug 2026) in Section 7, Outlook

Extending the present framework to quantify finite-depth corrections and to characterize landscapes away from the $2$-design regime is therefore an important direction for future work.

— A Representation-Theoretic Framework for Characterizing Barren Plateaus  (2609.04462 - Alcântara et al., 3 Sep 2026) in Discussion