# Identity-Paired Progressive Depth Training: When Trainability Persists Beyond Expressibility

arXiv: https://arxiv.org/abs/2607.16800 · HTML: https://mlsanity.com/paper/2607.16800/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 24 references that are in the map.
- First posted on arXiv 18 Jul 2026.

## Builds on (most cited first, 7 of 7)

- [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)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Understanding Progressive Training Through the Framework of Randomized Coordinate Descent](https://mlsanity.com/paper/2306.03626/) (2023)

## Abstract (arXiv)

Variational Quantum Algorithms (VQAs) are a leading paradigm for near-term quantum computing, yet their training suffers from sensitivity to circuit depth, initialization, and landscape pathologies such as barren plateaus. We study progressive depth training (PDT) -- a layerwise curriculum that trains a shallow circuit before appending new layers -- and identify a fundamental obstacle: fixed entangling gates (CNOTs) in hardware-efficient ans"atze cause initialization shock, an energy spike when new layers are added. We propose identity-paired progressive depth training (IP-PDT), which appends forward/inverse block pairs -- each consisting of a standard rotation+CNOT block followed by its reverse -- that compose to the identity at initialization. Because the adjacent CNOT rings cancel, the effective circuit retains only textita single entangling layer surrounded by textitoverparameterized local rotations. We prove a simple textitReachable Set Saturation Theorem: under this construction the variational manifold expands exactly once (when post-entangler rotations are first introduced) and then saturates; all subsequent depth increases provide pure overparameterization of single-qubit unitaries. Despite this saturation, progressive addition of rotation parameters can continue to improve optimization outcomes -- a phenomenon we term trainability beyond expressibility. We formalize IP-PDT as a continuation method on nested manifolds, prove monotone energy guarantees under an acceptance rule, and connect energy error to ground-state fidelity through spectral-gap inequalities. A detailed resource analysis shows that IP-PDT achieves lower total gate cost than both baselines by eliminating most CNOT gates.