Data quantum Fisher information predicts trainability in variational quantum algorithms

Abstract

Introduction: Variational quantum algorithms can show unstable trainability and uneven generalisation that are not fully explained by circuit depth or raw parameter count alone. We study the data quantum Fisher information matrix (DQFIM) as a data-dependent effective-capacity diagnostic for controlled supervised unitary-learning tasks. Materials and methods: We used exact-state, classically simulated, 4-qubit, supervised unitary-learning benchmarks with two ansatz families: a hardware-efficient ansatz (HEA) for the baseline trainability study and a symmetry-preserving ansatz for symmetry-controlled comparisons. The analysis covered four linked experiments: E0 trainability phase boundaries, E1 symmetry-controlled data-regime comparisons, E2 trainability versus generalisation, and E3 pre-training prediction of optimisation success. For the main analyses we used the support-basis DQFIM, 30 random seeds per main configuration, grouped cross-validation in the predictive benchmark, and additional diagnostics for threshold sensitivity, phase indeterminacy, Hamming-sector generalisation, and test set size sensitivity. Results: In E0, the empirical trainability boundary increased from M c = 32 at L = 1 to M c = 192 at L = 8 , and the support-basis DQFIM-predicted boundary matched the empirical boundary on the resolved scanned grid under the main analysis setting. In E1, the sector-preserving and sector-broken conditions showed no resolved empirical boundary shift on the scanned grid, with only mild low-L asymmetry in the DQFIM-predicted boundary. In E2, trainability and generalisation separated clearly: some regimes generalised well, while others reached near-zero training loss but retained high test loss. The phase diagnostic supported relative phase indeterminacy as a mechanism for failure on in-sector superpositions after basis-state training. In E3, parameter count alone was a weak predictor of optimisation success, with receiver operating characteristic area under the curve (ROC AUC) = 0.648 ; the structural baseline was stronger, with ROC AUC = 0.958 ; and the DQFIM-enhanced model performed best, with ROC AUC = 0.989 . Conclusions: In these small, idealised, classically simulated matched-family tasks, the support-basis DQFIM provides a useful data-dependent pre-training diagnostic of effective capacity on the retained data support. It tracks resolved trainability boundaries and contributes predictive information beyond raw parameter count and structural metadata. The conclusions remain scoped to controlled small-system simulations; larger-qubit, noisy, hardware-executed, and less-matched settings require further validation before claiming practical scalability.