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

2026-07-18

AI Summary

Variational quantum algorithms often suffer from optimization difficulties due to circuit depth sensitivity, initialization shocks, and barren plateaus. This work proposes Identity-Pair Progressive Depth Training (IP-PDT), a method that incrementally enhances expressivity by layer-wise insertion of forward/backward identity-pair blocks, preserving the overall identity mapping while avoiding abrupt energy shifts and maintaining trainability. Theoretical analysis reveals that trainability can improve even after expressivity saturation, leading to the formulation of a reachable-set saturation theorem and a formalization of IP-PDT as a continuous optimization framework over nested manifolds. Experiments demonstrate that IP-PDT significantly reduces CNOT gate counts, guarantees monotonic energy descent, and continues to enhance ground-state fidelity and optimization performance beyond expressivity saturation.

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.

Problem

• Variational Quantum Algorithms
• barren plateaus
• progressive depth training
• initialization shock
• trainability

Innovation

• Identity-Paired Progressive Depth Training
• Trainability beyond Expressibility
• Barren Plateaus
• Variational Quantum Algorithms
• Overparameterization