Decomposing Gradient Suppression in Barren Plateaus: Activity, Sign Organization, and Coupling

arXiv:2605.01319v2 Announce Type: replace Abstract: Barren plateaus (BPs) are conventionally characterized by suppressed gradient variance, but this aggregate description does not reveal how the loss of gradient signal is composed across Hamiltonian terms. We introduce a term-resolved framework that decomposes the second moment of the gradient exactly into pre-cancellation activity, sign organization, and their statistical coupling. A conditional random-sign model, which preserves termwise magnitudes while treating signs as independent and symmetric, provides exact references for organization and coupling. We apply the framework to a hardware-efficient ansatz (HEA) and a Hamiltonian variational ansatz (HVA) for the transverse-field and longitudinal-field Ising models. For the HEA, finite-size suppression of the second moment is carried almost entirely by decaying activity, accounting for 96.5-98.9% of the fitted log-slope across tested depths in both Hamiltonians, while organization shows no systematic scaling and coupling remains consistent with its random-sign reference. For the HVA, activity and organization instead grow with system size and contribute comparably to the second-moment scaling. A bias-corrected mean-gradient check is consistent with zero in every tested condition, so these results carry over approximately to the gradient variance. Microscopic sign-alignment analysis further shows sector-structured organization in the HVA, whereas the HEA exhibits only weak residual sign structure that does not accumulate into net signed organization. These patterns are reproduced across both Hamiltonians, providing a term-resolved characterization of BP-relevant gradient suppression beyond aggregate gradient statistics.

Abstract:Barren plateaus (BPs) are conventionally characterized by suppressed gradient variance, but this aggregate description does not reveal how the loss of gradient signal is composed across Hamiltonian terms. We introduce a term-resolved framework that decomposes the second moment of the gradient exactly into pre-cancellation activity, sign organization, and their statistical coupling. A conditional random-sign model, which preserves termwise magnitudes while treating signs as independent and symmetric, provides exact references for organization and coupling. We apply the framework to a hardware-efficient ansatz (HEA) and a Hamiltonian variational ansatz (HVA) for the transverse-field and longitudinal-field Ising models. For the HEA, finite-size suppression of the second moment is carried almost entirely by decaying activity, accounting for 96.5-98.9% of the fitted log-slope across tested depths in both Hamiltonians, while organization shows no systematic scaling and coupling remains consistent with its random-sign reference. For the HVA, activity and organization instead grow with system size and contribute comparably to the second-moment scaling. A bias-corrected mean-gradient check is consistent with zero in every tested condition, so these results carry over approximately to the gradient variance. Microscopic sign-alignment analysis further shows sector-structured organization in the HVA, whereas the HEA exhibits only weak residual sign structure that does not accumulate into net signed organization. These patterns are reproduced across both Hamiltonians, providing a term-resolved characterization of BP-relevant gradient suppression beyond aggregate gradient statistics.

Substantially revised and retitled (previously "Barren Plateaus as Destructive Interference: A Diagnostic Framework and Implications for Structured Ansatzes"). Adds an exact three-factor decomposition of the gradient second moment, a microscopic sign-alignment analysis, and replication on the longitudinal-field Ising model

Quantum Physics (quant-ph); Machine Learning (cs.LG)

arXiv:2605.01319 [quant-ph]

(or arXiv:2605.01319v2 [quant-ph] for this version)

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Decomposing Gradient Suppression in Barren Plateaus: Activity, Sign Organization, and Coupling

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