Horizon-Conditioned Proposals for Rare-Event Estimation with Variational and Hybrid Quantum Circuits

Estimating rare mission-time failures is difficult, as direct simulation may require enormous numbers of trajectories before a failure is observed. Importance sampling reduces this cost by changing the transition law, but requires an effective state- and horizon-dependent proposal. For a finite-horizon hitting problem, the optimal proposal is the Doob $h$-transform. We study a small reliability model for which the corresponding value function and the exact variance of any resulting proposal can both be computed, providing an exact way to evaluate learned proposals. We compare three horizon encodings crossed with two symmetry sectors on Born machines, together with hybrid dressed circuits and classical regressors, and study sensitivity to representation, initialization, a $\gamma$-retuned family of increasingly rare failure regimes, and finite measurement budgets. Within the architectures studied, the statewise-regressive models substantially outperform the Born-machine parameterizations. Across repeated initializations, additive time re-uploading remains the weakest tested horizon encoding, while a symmetry-restricted ansatz consistently improves median learning error despite using eight times fewer circuit parameters. Initialization strongly affects absolute performance, yet these broader design trends survive aggregation across seeds. Because both model classes ultimately produce normalized proposals, the observed gap cannot be attributed to normalization alone. The two quantum architectures also exhibit distinct sensitivity to finite-shot readout under the protocol studied, and across the $\gamma$-retuned family spanning $p_T=10^{-2}$ to $10^{-8}$ the classical-to-hybrid gap widens substantially. Together, these results show how an exactly evaluable rare-event benchmark can separate effects of representation, optimization, and measurement that are otherwise easy to conflate in quantum-learning comparisons.