Adaptive H-EFT-VA: A Provably Safe Trajectory Through the Trainability-Expressibility Landscape of Variational Quantum Algorithms

Barren-plateau results have established exponential gradient suppression as a widely cited obstacle to the scalability of variational quantum algorithms. When and whether these results extend to a given objective has been addressed through loss-specific arguments, but a general structural characterization has remained open. We show that the objective itself admits a fixed-observable representation if and only if the loss is affine in the measured statistics, thereby identifying the exact boundary of the standard concentration-based proof template. Existing transfer results for non-affine losses achieve this reduction under additional assumptions; our characterization implies that such a reduction is not structurally available for a class of non-affine objectives, placing them outside the automatic reach of the existing proof template. Beyond the affine regime, a chain-rule decomposition reveals three governing factors -- model responsivity, loss-side signal, and transmittance -- and induces a loss-class dichotomy: bounded-gradient losses inherit suppression, while amplification-capable losses can in principle counteract it. In the exponentially wide setting, both classes fail, but for different structural reasons. When the interface is instead designed at polynomial width -- exposing coarse-grained statistics rather than individual bitstring probabilities -- the exponential-dimensional obstruction is relaxed and the dichotomy plays a genuine role. In a numerical demonstration on a charge-conserving quantum system, the amplification-capable objective produces resolved gradients several orders of magnitude larger than affine and inheriting baselines at comparable shot budgets. Over the tested interval, its scaling trend is statistically distinguished from the exponential trend of both alternatives. The boundary is affine; what lies beyond it is a representation-design problem.

We study variational quantum distribution learning through a hierarchy of Walsh--Fourier approximations on the Boolean cube. At each level, a selected set of target Fourier coefficients defines a spectral truncation, which is projected onto the probability simplex and used as the target of a quantum circuit Born machine. Parameters learned at one level initialize the next through a warm-start map. We prove an end-to-end expected learning guarantee where the approximation term is determined by the omitted Fourier mass, while a normalized unbiased estimator yields an explicit statistical bound for empirical truncations. We then instantiate the abstract discrepancy conditions for total variation distance and relate the resulting distributional error to quantum-state fidelity. The total-variation specialization incurs the explicit factor $2^{n-1}$ under our normalized $\ell_2$ convention and is therefore informative only for sufficiently concentrated Fourier tails. The framework does not establish global trainability or eliminate barren plateaus; rather, it identifies the conditions under which low-to-high spectral training admits a approximation--estimation--optimization analysis.

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 ansätze 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 a single entangling layer surrounded by overparameterized local rotations. We prove a simple Reachable 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.