Demonstrating Quadratic Monte Carlo Speedup via Quantum Amplitude Estimation: Nuclear Engineering Examples

We demonstrate quantum amplitude estimation (QAE) as a route to quadratic speedup for Monte Carlo-type expectation values in nuclear engineering. Using QPE-based QAE, we study two examples: a discrete fission-neutron-yield expectation and a U-238 resonance integral under a $1\mathrm{/}E$ slowing-down spectrum. The toy problem is implemented as a gate-level Qiskit circuit, while the resonance-integral example is simulated through an exact eigendecomposition of the Grover operator to avoid state-preparation decomposition bottlenecks. In both cases, the squared error scales as $O(1\mathrm{/}{T}^2)$ with the number of oracle calls $T$, compared with the classical Monte Carlo scaling $O(1\mathrm{/}N)$. For the U-238 example, QAE recovers the resonance integral to approximately $0.03$ relative error with $m=14$ phase-estimation qubits.