# From Trainability Diagnostics to Optimization Claims: Boundaries and Controls in Variational Quantum Optimization

arXiv: https://arxiv.org/abs/2609.21243 · HTML: https://mlsanity.com/paper/2609.21243/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 28 references that are in the map.
- First posted on arXiv 18 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Effect of barren plateaus on gradient-free optimization](https://mlsanity.com/paper/2011.12245/) (2020)
- [ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2408.11104/) (2024)
- [Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus](https://mlsanity.com/paper/2102.01828/) (2021)
- [Variational quantum simulation: a case study for understanding warm starts](https://mlsanity.com/paper/2404.10044/) (2024)
- [Decomposing Gradient Suppression in Barren Plateaus: Activity, Sign Organization, and Coupling](https://mlsanity.com/paper/2605.01319/) (2026)

## Abstract (arXiv)

Barren plateau diagnostics characterize whether gradient signal remains available for training, but surviving signal need not translate into successful optimization. We study this trainability--optimization gap at the level of optimizer steps. Treating coefficient-weighted Hamiltonian-term gradients as task-like components, we introduce step-level diagnostics and derive an exact bridge between signed termwise organization, directional activity, and first-order descent. Resolving this bridge into standard first-order geometry shows that the apparent organization--activity factors are not independent optimization axes and that, at fixed state and update norm, the raw gradient maximizes first-order descent of the summed objective. We compare vanilla gradient descent, a deterministic Hamiltonian-term PCGrad variant, and probe-gated LSO-PCGrad on transverse-field Ising model instances with hardware-efficient and Hamiltonian variational ansatzes, together with matched controls for update norm and probe budget. Blind projection can improve an organization diagnostic while worsening final energy and first-order predictability. After conditioning on standard first-order geometry, residual term-space composition shows no reproducible material incremental association with realized descent, while optimizer-relative update norm shows positive material associations in some settings without cross-regime reproducibility. Matched controls provide no resolved final-energy benefit attributable to the projected direction, and the improvement of LSO-PCGrad is more consistent with probe-based search and step-norm adaptation than with Hamiltonian-term projection itself. These results show that gradient-structure diagnostics can characterize trainability and update geometry without serving as standalone evidence of optimization benefit, which requires controls matched on update norm and search budget.