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

Abstract: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 \emph{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 ansätze cause \emph{initialization shock}, an energy spike when new layers are added. We propose \emph{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 \textit{a single entangling layer} surrounded by \textit{overparameterized local rotations}. We prove a simple \textit{Reachable Set Saturation Theorem}: under this construction the variational manifold expands exactly once (when post-entangler rotations are first introduced) and then \emph{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 \emph{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.

Quantum Physics (quant-ph); Machine Learning (cs.LG); Optimization and Control (math.OC)

arXiv:2607.16800 [quant-ph]

(or arXiv:2607.16800v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.16800 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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