Quantum Amplitude Estimation for Catastrophe Insurance Tail-Risk Pricing: Empirical Convergence and NISQ Noise Analysis

Abstract

Classical Monte Carlo methods for pricing catastrophe insurance tail risk converge at order reciprocal root N, requiring large simulation budgets to resolve upper-tail percentiles of the loss distribution. This sample-sparsity problem can lead to AI models trained on impoverished tail data, producing poorly calibrated risk estimates where insolvency risk is greatest. Quantum Amplitude Estimation (QAE), following Montanaro, achieves convergence approaching order reciprocal N in oracle queries - a quadratic speedup that, at scale, would enable high-resolution tail estimation within practical budgets. We validate this advantage empirically using a Qiskit Aer simulator with genuine Grover amplification. A complete pipeline encodes fitted lognormal catastrophe distributions into quantum oracles via amplitude encoding, producing small readout probabilities that enable safe Grover amplification with up to k=16 iterations. Seven experiments on synthetic and real (NOAA Storm Events, 58,028 records) data yield three main findings: an oracle-model advantage, that strong classical baselines win when analytical access is available, and that discretisation, not estimation, is the current bottleneck.

Figures (9)

• Figure 1: Experiment 1: log--log convergence of RMSE vs total oracle queries for quantum AE (Grover-amplified, $k=0\ldots6$) and classical MC on the same 8-bin discretisation. Dashed lines show the $O(1/\sqrt{N})$ and $O(1/N)$ reference slopes.
• Figure 2: QAE RMSE under increasing NISQ noise at $k=3$ Grover iterations. The noiseless baseline (RMSE $=$ $32) degrades to $\sim$$2,100 under even low noise---a $66\times$ increase, driven by the $\sim$700-gate circuit depth.
• Figure 3: Experiment 3: estimation error vs exact-on-bins (left) comparing classical MC on the same bins and quantum Grover, with speedup ratios annotated; discretisation error vs classical continuous accuracy (right) providing end-to-end context.
• Figure 4: Experiment 4A: log--log convergence of RMSE vs total oracle queries on real NOAA data ($P(\ket{1}) = 0.0040$, $k_{\max} = 11$). Quantum AE converges faster than classical MC on the same bins, consistent with the synthetic results.
• Figure 5: Experiment 4B: RMSE by tail percentile on real NOAA data. The 90th-percentile bar is dominated by pathological equal-width discretisation error ($146K). At the 95th and 97th percentiles, the quantum advantage is 2--2.5$\times$ in the oracle-model comparison (Table \ref{['tab:exp4b']}).