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

Alexis Kirke

stat.APMar 10, 2026

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

Authors: Alexis Kirke

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 & tables

Filename	Year
StormEvents_details-ftp_v1.0_d2020_c20260116.csv.gz	2020
StormEvents_details-ftp_v1.0_d2021_c20250520.csv.gz	2021
StormEvents_details-ftp_v1.0_d2022_c20250721.csv.gz	2022
StormEvents_details-ftp_v1.0_d2023_c20260116.csv.gz	2023
StormEvents_details-ftp_v1.0_d2024_c20260116.csv.gz	2024

Table 1: NOAA Storm Events files used in Experiments 4 and 7. Retrieved 2026-02-27 from ncei.noaa.gov.

$k$	Queries	Q (vs bins)	$C_{\text{bins}}$ (vs bins)	Speedup	$C_{\text{cont}}$ (vs anal.)
0	1,000	930	782	0.8 $\times$	948
1	3,000	283	382	1.4 $\times$	482
2	5,000	180	303	1.7 $\times$	287
3	7,000	124	248	2.0 $\times$	352
4	9,000	106	139	1.3 $\times$	235
5	11,000	87	177	2.0 $\times$	263

Table 2: Experiment 1: error decomposition at the 95th-percentile threshold (

P(\ket{1})=0.0066

k_{\max}=9

n_{\text{reps}}=30

, 1,000 shots/run). Analytic

E[\text{excess}]=\

2{,}834

; exact-on-bins

=$2{,}330

; disc. error

=$505

. Speedup

=C_{\text{bins}}/\text{Q}

; values

{>}1$ indicate quantum is better.

Figure 1: Experiment 1: log–log convergence of RMSE vs total oracle queries for quantum AE (Grover-amplified, $k=0\ldots 6$ ) and classical MC on the same 8-bin discretisation. Dashed lines show the $O(1/\sqrt{N})$ and $O(1/N)$ reference slopes.

Noise Level	$p_{1q}$ / $p_{2q}$	RMSE ($)	Mean Est. ($)	Std. ($)	Bias ($)
Noiseless	0 / 0	32	2,332	32	+2
Low	0.001 / 0.01	2,094	4,423	50	+2,093
Medium	0.005 / 0.05	2,115	4,444	62	+2,114
High	0.01 / 0.10	2,115	4,444	67	+2,114

Table 3: Experiment 2: QAE accuracy under NISQ noise (

k=3

, 8,192 shots,

n_{\text{reps}}=20

). Exact

E[\text{excess}]=\

2{,}330$ .

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 2: QAE RMSE under increasing NISQ noise at $k=3$ Grover iterations. The noiseless baseline (RMSE $=$ $32) degrades to$ \sim2,100 under even low noise—a $66\times$ increase, driven by the $\sim$ 700-gate circuit depth.

Pctl	$M$	Anal.	Bins	Disc. err	Q / $C_{\text{bins}}$ / Speedup	$C_{\text{cont}}$
90th	230,876	9,629	9,441	188	319 / 486 / 1.5 $\times$	598
95th	362,700	2,834	2,330	505	93 / 193 / 2.1 $\times$	278
97th	508,392	900	498	401	29 / 74 / 2.5 $\times$	178

Table 4: Experiment 3: error decomposition at tail percentiles (

n_{\text{reps}}=30

\approx

8,192 queries). Q and

C_{\text{bins}}

are RMSE vs exact-on-bins;

C_{\text{cont}}

is RMSE vs analytic truth. Disc. err

=|\text{bins}-\text{analytic}|

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.

$k$	Queries	Q (vs bins)	$C_{\text{bins}}$ (vs bins)	Speedup	$C_{\text{cont}}$ (vs anal.)
0	1,000	8,111	5,189	0.6 $\times$	12,285
1	3,000	2,161	2,464	1.1 $\times$	6,667
2	5,000	1,160	2,087	1.8 $\times$	4,982
3	7,000	929	1,796	1.9 $\times$	5,752
4	9,000	705	1,188	1.7 $\times$	3,009
5	11,000	548	1,196	2.2 $\times$	4,029

Table 5: Experiment 4A: RMSE (

) on real NOAA data (95th pctl,

n_{\text{reps}}=30

, 1,000 shots/run). Analytic

E[\text{excess}]=$21{,}845

; exact-on-bins

=$14{,}121

; disc. error

=$7{,}724$ .

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.

Pctl	$M$ ($)	Analytic	Exact bins	Disc. err	$C_{\text{cont}}$	$C_{\text{bins}}$	Q ( $k$ )	Speedup
90th †	100,000	39,655	186,273	146,619	4,445	1,944	4,228 (3)	0.5 $\times$
95th	475,650	21,845	14,121	7,724	4,508	1,413	656 (6)	2.2 $\times$
97th	1,000,000	14,434	6,534	7,900	3,403	999	406 (6)	2.5 $\times$

Table 6: Experiment 4B: error decomposition on real NOAA data (

n_{\text{reps}}=30

\approx

8,192 total queries).

C_{\text{cont}}

samples the continuous lognormal (RMSE vs analytic);

C_{\text{bins}}

and Q sample from the same 8-bin discretisation (RMSE vs exact-on-bins). † Pathological: excluded from headline results due to catastrophic equal-width binning (see text).

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 6 ). — 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 6 ).

Pctl	Analytic	Exact bins	Naive MC	CT	IS	QMC	Cl. bins	QAE ( $k$ )
90th	9,629	9,441	518	131	158	37	425	266 (5)
95th	2,834	2,330	369	32	45	37	527	484 (9)
97th	900	498	169	10	14	37	436	387 (15)

Table 7: Experiment 5: RMSE (

) vs analytic truth at budget

B=8{,}192

(synthetic data, 50 reps). “Exact bins” is the deterministic sum

\sum p_{i}\max(0,x_{i}-M)$ —a non-sampling reference requiring zero queries.

Panel of Figure 6 — Figure 6: Experiment 5: Left: all methods at $B=8{,}192$ across three percentiles (synthetic data). CT and IS dominate because they sample the continuous fitted distribution, avoiding discretisation error entirely. Right: RMSE vs budget at the 95th percentile; CT and IS converge to zero while quantum and classical-on-bins flatten at the discretisation floor ( $\approx$ $505).

$n$	Bins	Disc. err ($)	Q-RMSE ${}_{\text{est}}$ ($)	C-RMSE ${}_{\text{est}}$ ($)	$k$	SP 2Q gates	Oracle depth
3	8	505	91	349	9	52	388
4	16	578	115	282	9	212	1,063
5	32	557	136	306	9	596	2,434
6	64	564	98	261	9	1,876	5,958
7	128	566	103	304	9	5,460	21,150
8	256	567	145	326	9	15,700	63,238

Table 8: Experiment 6: error decomposition across qubit counts (95th pctl,

B=4{,}000

, 50 reps). Circuit metrics are from transpilation to

\{CX,R_{z},SX,X\}

Figure 7: Experiment 6: error decomposition (left) and total RMSE vs analytic truth (right). Discretisation error dominates and remains $\sim$ $505–$578 across all qubit counts because the bin range (0.1st–99.9th percentile) truncates the upper tail identically. — Figure 7: Experiment 6: error decomposition (left) and total RMSE vs analytic truth (right). Discretisation error dominates and remains $\sim$ $505–$ 578 across all qubit counts because the bin range (0.1st–99.9th percentile) truncates the upper tail identically.

Figure 8: Experiment 6: circuit resource scaling. State-preparation two-qubit gates grow as $O(2^{n})$ ; at $n=7$ , a single oracle call requires 21,150 depth and 5,460 CX gates.

Pctl	Exact ($)	Naive MC ($)	Class. bins ($)	QAE ( $k$ ) ($)	Speedup
90th	1,683,761	608,949	57,351	24,076 (1)	2.4 $\times$
95th	1,634,141	622,976	56,930	27,156 (1)	2.1 $\times$
97th	1,564,879	862,260	48,776	30,632 (1)	1.6 $\times$

Table 9: Experiment 7: RMSE (

) vs exact-on-bins at

B=8{,}000$ (NOAA empirical PMF, 50 reps, quantile-binned). The exact-on-bins value is shown for reference; it requires zero samples.

Figure 9: Experiment 7: RMSE on the empirical NOAA PMF (no parametric fit). QAE achieves 1.6–2.4 $\times$ lower RMSE than classical MC on the same bins. CT and IS offer no direct advantage without extra modelling.

Binning	$n$	Bins	Disc. err ($)	Q-RMSE ($)	C-RMSE ($)	Speedup	$k_{\max}$
Equal-width	3	8	505	101	345	3.4 $\times$	9
Log-spaced	3	8	807	185	520	2.8 $\times$	8
Equal-width	4	16	578	116	275	2.4 $\times$	9
Log-spaced	4	16	117	52	391	7.5 $\times$	11
Equal-width	5	32	557	126	302	2.4 $\times$	9
Log-spaced	5	32	54	67	278	4.1 $\times$	12

Table 10: Equal-width vs. log-spaced binning (95th pctl,

B=4{,}000

, 30 reps). Log-spaced bins extend to the 99.99th percentile with geometric midpoints.

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Heavy-Tailed Distributions Rare-Event Estimation

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Financial crashes, cascading failures in infrastructure, and critical errors in AI systems are frequently triggered by events that occur with extremely small probability. Efficiently discovering and sampling events with probability below a threshold is therefore of critical interest. Yet this task is highly non-trivial using existing classical or quantum methods. Being rare, such events require an immense sampling overhead to collect sufficient data samples. Moreover, because the rare events are not known in advance, they cannot be flagged for amplification using standard techniques. Here, we introduce a quantum algorithm for rare-event discovery and sampling without first learning which events are rare. The algorithm achieves the optimal quantum scaling with the rarity threshold. We further demonstrate that this can achieve a quadratic speedup for heavy-tailed systems whose tail has nonvanishing total mass, and translates into a robust polynomial speedup for stationary stochastic processes, with the exponent determined by its entropy-rate structure.

Abstract

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