Characterization of the tightness of the trainability upper bound

Determine whether the upper bound on the expected gradient variance of the single-layer Quantum Simplified Graph Convolutional Network, scaling as \(\Theta(C^{-1})\), is tight by identifying an ansatz that maximizes the expected variance.

Background

The paper derives an upper bound of order Θ(C−1)\Theta(C^{-1}) for the expected gradient variance, where CC is the number of node features. It also explains that establishing tightness requires finding a parameterized quantum ansatz that attains the maximum possible expected variance. This question is left unresolved.

References

Finding whether this bound is tight is a work in progress, as one needs to find the right ansatz that maximizes the expected variance.

— Quantum Graph Convolutional Networks: Implementation and Trainability Analysis  (2609.19983 - Sein et al., 17 Sep 2026) in Appendix, Section "Bounds for the expected variance," subsection "Upper bound"