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arXiv:2607.10772

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

Jilang Miao, Miaomiao Jin, Akira Sone

quant-phphysics.comp-phSubmitted Jul 12, 2026• Announced Jul 14, 2026• 0 citations
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Abstract

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 $1/E$ 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 $O(1/T^2)$ with the number of oracle calls $T$, compared with the classical Monte Carlo scaling $O(1/N)$. For the U-238 example, QAE recovers the resonance integral to approximately $0.03%$ relative error with $m=14$ phase-estimation qubits.

AI Summary

The paper studies QPE-based quantum amplitude estimation (QAE) for two nuclear-engineering expectation values, comparing estimation error against classical Monte Carlo sampling. It encodes a probability distribution and bounded scoring function into a state-preparation unitary, then estimates the associated Grover eigenphase using $m$ phase-estimation qubits and $T=2^m-1$ Grover-operator calls. The first example is a four-outcome fission-neutron-yield model with exact mean $\bar\nu=2$. Qiskit statevector simulations use $m=2$–$16$ and 8,192 shots per setting, decoding the most frequent phase-register outcome. The second example estimates a U-238 resonance integral around the 6.674 eV resonance, discretized into 256 energy bins under a $1/E$ spectrum. Rather than decomposing its state-preparation circuit, this example constructs a $512\times512$ Grover matrix and simulates QPE through exact eigendecomposition and Fourier transforms. Both examples report decoded squared errors consistent with $O(T^{-2})$, compared with classical Monte Carlo MSE $O(N^{-1})$. At $m=14$, the resonance estimate is 120.65 b versus a reference value of 120.69 b, approximately 0.03% relative error. The paper compares its squared amplitude error with approximately 49.4 million classical samples, yielding a nominal sample-to-oracle-call ratio of 3,016; each QAE setting nevertheless uses 8,192 shots. The stated implementation limitation is the cost of generic state-preparation decomposition, which prevents gate-level simulation of the larger example. The eigendecomposition surrogate reproduces the ideal measurement distribution but does not provide a hardware-ready circuit. Quantum-arithmetic state preparation, larger discretizations, multiple resonances, and iterative QAE are left for future work.

Some signs of AI-generated textOverclaimsIncrementalSmall numerics onlyMissing comparison

Central claim may not hold: The paper labels modal squared-error curves as MSE without establishing the mean squared error of its finite-shot mode estimator. The claimed approximately 3016-fold numerical speedup omits 8192 repetitions, which increase the implemented QAE cost to approximately 134.2 million Grover calls versus the quoted 49.4 million classical samples. The stated resonance parameters and discretization yield C_phys approximately 1587.86 b and μ approximately 0.076011, not the reported 1578 b and 0.07648.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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4 pages,3 figures, submitted to 2026 ANS Annual Conference

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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/E1/E1/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/T2)O(1/T^2)O(1/T2) with the number of oracle calls TTT, compared with the classical Monte Carlo scaling O(1/N)O(1/N)O(1/N). For the U-238 example, QAE recovers the resonance integral to approximately 0.030.03%0.03 relative error with m=14m=14m=14 phase-estimation qubits.
The paper studies QPE-based quantum amplitude estimation (QAE) for two nuclear-engineering expectation values, comparing estimation error against classical Monte Carlo sampling. It encodes a probability distribution and bounded scoring function into a state-preparation unitary, then estimates the associated Grover eigenphase using mmm phase-estimation qubits and T=2m−1T=2^m-1T=2m−1 Grover-operator calls.
The first example is a four-outcome fission-neutron-yield model with exact mean νˉ=2\bar\nu=2νˉ=2. Qiskit statevector simulations use m=2m=2m=2–161616 and 8,192 shots per setting, decoding the most frequent phase-register outcome. The second example estimates a U-238 resonance integral around the 6.674 eV resonance, discretized into 256 energy bins under a 1/E1/E1/E spectrum. Rather than decomposing its state-preparation circuit, this example constructs a 512×512512\times512512×512 Grover matrix and simulates QPE through exact eigendecomposition and Fourier transforms.
Both examples report decoded squared errors consistent with O(T−2)O(T^{-2})O(T−2), compared with classical Monte Carlo MSE O(N−1)O(N^{-1})O(N−1). At m=14m=14m=14, the resonance estimate is 120.65 b versus a reference value of 120.69 b, approximately 0.03% relative error. The paper compares its squared amplitude error with approximately 49.4 million classical samples, yielding a nominal sample-to-oracle-call ratio of 3,016; each QAE setting nevertheless uses 8,192 shots.
The stated implementation limitation is the cost of generic state-preparation decomposition, which prevents gate-level simulation of the larger example. The eigendecomposition surrogate reproduces the ideal measurement distribution but does not provide a hardware-ready circuit. Quantum-arithmetic state preparation, larger discretizations, multiple resonances, and iterative QAE are left for future work.
We introduce a variant of Quantum Amplitude Estimation (QAE), called Iterative QAE (IQAE), which does not rely on Quantum Phase Estimation (QPE) but is only based on Grover’s Algorithm, which reduces the required number of qubits and gates. We provide a rigorous analysis of IQAE and prove that it achieves a quadratic speedup up to a double-logarithmic factor compared to classical Monte Carlo simulation with provably small constant overhead. Furthermore, we show with an empirical study that our algorithm outperforms other known QAE variants without QPE, some even by orders of magnitude, i.e., our algorithm requires significantly fewer samples to achieve the same estimation accuracy and confidence level.
Abstract We introduce a variant of Quantum Amplitude Estimation (QAE) , called Iterative QAE (IQAE), which does not rely on Quantum Phase Estimation (QPE) but is only based on Grover’s Algorithm , which reduces the required number of qubits and gates. We provide a rigorous analysis of IQAE and prove that it achieves a quadratic speedup up to a double-logarithmic factor compared to classical Monte Carlo simulation with provably small constant overhead. Furthermore, we show with an empirical study that our algorithm outperforms other known QAE variants without QPE, some even by orders of magnitude, i.e., our algorithm requires significantly fewer samples to achieve the same estimation accuracy and confidence level.
Quantum amplitude estimation (QAE) can achieve a quadratic speedup for applications classically solved by Monte Carlo simulation. A key requirement to realize this advantage is efficient state preparation. If state preparation is too expensive, it can diminish the quantum advantage. Preparing arbitrary quantum states has exponential complexity with respect to the number of qubits, and thus, is not applicable. Currently known efficient techniques require problems based on log-concave probability distributions, involve learning an unknown distribution from empirical data, or fully rely on quantum arithmetic. In this paper, we introduce an approach to simplify state preparation, together with a circuit optimization technique, both of which can help reduce the circuit complexity for QAE state preparation significantly. We demonstrate the introduced techniques for a numerical integration example on real quantum hardware, as well as for option pricing under the Heston model, i.e., based on a stochastic volatility process, using simulation.
Option pricing is a fundamental problem in quantitative finance, where the accuracy and efficiency of valuation methods directly affect investment and risk management decisions. While the Black-Scholes model provides a closed-form solution for European options, numerical approaches such as Monte Carlo simulation are widely used when analytical methods are intractable. However, Monte Carlo suffers from slow convergence with complexity O(1/ε2).Quantum Amplitude Estimation (QAE), an algorithm in quantum computing, promises a quadratic speedup with complexity O(1/ε). This paper implements QAE, specifically Iterative Amplitude Estimation (IAE), for European call option pricing and compares its performance against Monte Carlo simulation and the Black-Scholes model. Experiments are conducted on both Qiskit simulators and IBM Quantum hardware (ibm_brisbane).Results show that QAE achieves estimates close to Monte Carlo and analytical benchmarks, with mean payoff deviations of less than 0.01. While QAE produces wider confidence intervals due to noise and discretization, the findings demonstrate its potential scalability advantage for future quantum finance applications.
The Quantum Amplitude Estimation (QAE) algorithm is a major quantum algorithm designed to achieve a quadratic speed-up. Until fault-tolerant quantum computing is achieved, being competitive over classical Monte Carlo (MC) remains elusive. Alternative methods have been developed so as to require fewer resources while maintaining an advantageous theoretical scaling. We compared the standard QAE algorithm with two Noisy Intermediate-Scale Quantum (NISQ)-friendly versions of QAE on a numerical integration task, with the Monte Carlo technique of Metropolis–Hastings as a classical benchmark. The algorithms were evaluated in terms of the estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the solutions, respectively. The effectiveness of the two QAE alternatives was tested on an 11-qubit trapped-ion quantum computer in order to verify which solution can first provide a speed-up in the integral estimation problems. We concluded that an alternative approach is preferable with respect to employing the phase estimation routine. Indeed, the Maximum Likelihood estimation guaranteed the best trade-off between the length of the quantum circuits and the precision in the integral estimation, as well as greater resistance to noise.
Quantum Amplitude Estimation (QAE) is an important quantum algorithm that has the potential to quadratically speed up Monte Carlo based calculations.In this paper, we present a variant of the QAE without Phase Estimation Algorithm called Iterative Refinement QAE (IRQAE). IRQAE can refine the current estimation to a more accurate estimation iteratively, hence it can provide an estimation with arbitrary required accuracy ∊. The key idea of IRQAE is to use a rotation gate to create a quantum state for samplings with the current estimation. Using this idea, we show that IRQAE can provide a highly accurate estimation with lower classical computational complexity and with the same quantum computational complexity compared to state-of-the-art QAEs without phase estimation using numerical experiments. We prove that the computational complexity of IRQAE of the quantum part is O(1/∊) and the classical one is O(1/∊). The quantum cost gives a quadratic advantage over that of the classical Monte Carlo simulation.