Identity-Paired Progressive Depth Training: When Trainability Persists Beyond Expressibility
Machine Learning
Summary
The authors study how to train quantum circuits layer by layer to avoid sudden performance drops, a problem they call "initialization shock." They propose a method called identity-paired progressive depth training (IP-PDT) that adds pairs of operations which cancel out at first, keeping the circuit stable and easier to optimize. They prove that after a certain point, adding more layers doesn't increase the circuit's ability to represent new states but still helps make training better. Their method also reduces the number of costly entangling gates required, making training more efficient.
Authors
Athanasios Hadjidimoulas, Tirthak Patel, Anastasios Kyrillidis
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.