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

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.