Speaker
Description
Barren-plateau theorems establish exponential gradient suppression for variational objectives of the fixed-observable form $\operatorname{Tr}[H \rho(\theta)]$. We give an exact structural characterization of when this form is available: relative to a chosen measurement interface — the set of expectation values the loss may depend on — a fixed-observable representation exists on an explored region if and only if the loss is affine in the exposed statistics. Non-affine objectives (divergences, likelihoods, risk functionals) therefore lie outside the automatic reach of the concentration-based proof template. Beyond this boundary, a chain-rule decomposition identifies 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 need not. Both classes fail at exponentially wide interfaces; polynomial width restores the dichotomy. On a charge-conserving system with a block-charge interface, a negative log-likelihood head resolves gradients roughly $10^4$ times larger than affine baseline at matched shot cost ($n_{qubits}$ = 8–24), with a scaling trend statistically separated from the exponential class — while the required shot budget remains exponential for all heads, localizing the remaining obstruction in the responsivity of the exposed statistics. We state the constructive open problem as the Polynomially-Barren & Just-Right (PB&J) hypothesis (arXiv:2604.18846). The boundary is affine; what lies beyond it is a representation-design problem.
| Topic | Theoretical foundations of quantum learning |
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