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arXiv:2609.25992

Maximum-Likelihood Amplitude Estimation for Quantum Monte Carlo Integration on Trapped-Ion Hardware: Convergence, Noise-Floor Saturation, Depth-Dependent Bias

Aditi Lal, Alex Khan, Rut Lineswala, Abhishek Chopra

quant-phSubmitted Sep 22, 2026• Announced Sep 23, 2026• 0 citations
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Abstract

Quantum Amplitude Estimation promises to accelerate Monte Carlo integration by replacing the classical $\varepsilon \propto N^{-1/2}$ sampling law with the Heisenberg-limited $\varepsilon \propto N^{-1}$ scaling, but whether any part of that advantage survives on present-day hardware is an empirical question. We report a systematic experimental benchmark of a Maximum-Likelihood Amplitude Estimation QMC pipeline on IonQ trapped-ion processors, comprising 60 hardware trials on IonQ Forte-1 across five settings(5 & 9 qubits, 100-200 shots per depth, linear & exponential amplification schedules, ~113 device-hours), together with 100 trials on IonQ Aria-1 & Forte-1 noise models at 5, 9 & 19 qubits. The pipeline combines controlled-$R_y$ rotational state preparation, Grover-based amplitude amplification and maximum-likelihood inference and is benchmarked against $b_{\max}^{-1}\!\int_0^{b_{\max}}\!\sin^2\!x\,\mathrm{d}x$ with $b_{\max}=π/5$ ,given results emerge. First, the noiseless limit it attains $\varepsilon \propto N^{-0.88}$, close to the Heisenberg-scaling reference, reducing the error 54-74x over a 133x increase in oracle calls. Second, scaling does not survive on hardware. Every noisy backend saturates at an error floor of (2-5)x 10^{-2}, with fitted exponents of 0.04-0.15 on Forte-1 linear schedules and $α\le\!0$ on the vendor noise models: beyond the optimal shallow depth, however, deeper amplification provides no sustained accuracy gain and instead returns to the noise floor. Forte-1 hardware also outperformed IonQ's own Aria-1 noise model at every depth $m \ge 2$, indicating that vendor noise models are bad predictors of MLAE accuracy. The practical implication for UQ is that on current devices the useful operating point is a shallow, schedule-tuned amplification depth chosen so that the deepest circuit lands near p~1/2 not the deepest schedule the coherence budget allows.

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7 pages 3 figures

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Quantum Amplitude Estimation promises to accelerate Monte Carlo integration by replacing the classical ε∝N−1/2\varepsilon \propto N^{-1/2}ε∝N−1/2 sampling law with the Heisenberg-limited ε∝N−1\varepsilon \propto N^{-1}ε∝N−1 scaling, but whether any part of that advantage survives on present-day hardware is an empirical question. We report a systematic experimental benchmark of a Maximum-Likelihood Amplitude Estimation QMC pipeline on IonQ trapped-ion processors, comprising 60 hardware trials on IonQ Forte-1 across five settings(5 & 9 qubits, 100-200 shots per depth, linear & exponential amplification schedules, 113 device-hours), together with 100 trials on IonQ Aria-1 & Forte-1 noise models at 5, 9 & 19 qubits. The pipeline combines controlled-RyR_yRy​ rotational state preparation, Grover-based amplitude amplification and maximum-likelihood inference and is benchmarked against bmax⁡−1 ⁣∫0bmax⁡ ⁣sin⁡2 ⁣x dxb_{\max}^{-1}\!\int_0^{b_{\max}}\!\sin^2\!x\,\mathrm{d}xbmax−1​∫0bmax​​sin2xdx with bmax⁡=π/5b_{\max}=π/5bmax​=π/5 ,given results emerge. First, the noiseless limit it attains ε∝N−0.88\varepsilon \propto N^{-0.88}ε∝N−0.88, close to the Heisenberg-scaling reference, reducing the error 54-74x over a 133x increase in oracle calls. Second, scaling does not survive on hardware. Every noisy backend saturates at an error floor of (2-5)x 10^{-2}, with fitted exponents of 0.04-0.15 on Forte-1 linear schedules and α≤ ⁣0α\le\!0α≤0 on the vendor noise models: beyond the optimal shallow depth, however, deeper amplification provides no sustained accuracy gain and instead returns to the noise floor. Forte-1 hardware also outperformed IonQ's own Aria-1 noise model at every depth m≥2m \ge 2m≥2, indicating that vendor noise models are bad predictors of MLAE accuracy. The practical implication for UQ is that on current devices the useful operating point is a shallow, schedule-tuned amplification depth chosen so that the deepest circuit lands near p1/2 not the deepest schedule the coherence budget allows.
Amplitude estimation is a fundamental quantum algorithmic primitive that enables quantum computers to achieve quadratic speedups for a large class of statistical estimation problems, including Monte Carlo methods. The main drawback from the perspective of near term hardware implementations is that the amplitude estimation algorithm requires very deep quantum circuits. Recent works have succeeded in somewhat reducing the necessary resources for such algorithms, by trading off some of the speedup for lower depth circuits, but high quality qubits are still needed for demonstrating such algorithms. Here, we report the results of an experimental demonstration of amplitude estimation on a state-of-the-art trapped ion quantum computer. The amplitude estimation algorithms were used to estimate the inner product of randomly chosen four-dimensional unit vectors, and were based on the maximum likelihood estimation (MLE) and the Chinese remainder theorem (CRT) techniques. Significant improvements in accuracy were observed for the MLE based approach when deeper quantum circuits were taken into account, including circuits with more than ninety two-qubit gates and depth sixty, achieving a mean additive estimation error on the order of 10−210^{-2}10−2. The CRT based approach was found to provide accurate estimates for many of the data points but was less robust against noise on average. Last, we analyze two more amplitude estimation algorithms that take into account the specifics of the hardware noise to further improve the results.
We propose to perform amplitude estimation with the help of constant-depth quantum circuits that variationally approximate states during amplitude amplification. In the context of Monte Carlo (MC) integration, we numerically show that shallow circuits can accurately approximate many amplitude amplification steps. We combine the variational approach with maximum likelihood amplitude estimation [Y. Suzuki et al., Quantum Inf. Process. 19, 75 (2020)] in variational quantum amplitude estimation (VQAE). VQAE typically has larger computational requirements than classical MC sampling. To reduce the variational cost, we propose adaptive VQAE and numerically show in 6 to 12 qubit simulations that it can outperform classical MC sampling.
Quantum Amplitude Estimation (QAE) -- a technique by which the amplitude of a given quantum state can be estimated with quadratically fewer queries than by standard sampling -- is a key sub-routine in several important quantum algorithms, including Grover search and Quantum Monte-Carlo methods. An obstacle to implementing QAE in near-term noisy intermediate-scale quantum (NISQ) devices has been the need to perform Quantum Phase Estimation (QPE) -- a costly procedure -- as a sub-routine. This impediment was lifted with various QPE-free methods of QAE, wherein Grover queries of varying depths / powers (often according to a "schedule") are followed immediately by measurements and classical post-processing techniques like maximum likelihood estimation (MLE). Existing analyses as to the optimality of various query schedules in these QPE-free QAE schemes have hitherto assumed noise-free systems. In this work, we analyse QPE-free QAE under common noise models that may afflict NISQ devices and report on the optimality of various query schedules in the noisy regime. We demonstrate that, given an accurate noise characterization of one's system, one must choose a schedule that balances the trade-off between the greater ideal performance achieved by higher-depth circuits, and the correspondingly greater accumulation of noise-induced error.
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.
We develop a class of amplitude estimation algorithms with tunable circuit depth MMM and circuit repetitions NNN, requiring neither ancilla qubits nor controlled Grover operations. For additive error εεε in the Grover angle, they attain the nearly optimal query-depth tradeoff M2N∈O~(ε−2)M^2N\in\widetilde{O}(ε^{-2})M2N∈O(ε−2) uniformly over λ∈[0,π/2]λ\in[0,π/2]λ∈[0,π/2], spanning the full range from classical sampling at M=1M=1M=1 to the Heisenberg limit at M=Θ(ε−1)M=Θ(ε^{-1})M=Θ(ε−1). While prior depth-tunable work establishes comparable angle-accuracy guarantees only away from the boundaries λ→0λ\to0λ→0 or π/2π/2π/2 or at discrete depth tradeoffs, our guarantee extends to both boundaries, so the quantum speedup persists there rather than degrading to classical sampling. Numerical experiments confirm the predicted uniform angle accuracy and show low overhead in practice, making them strong candidates for practical amplitude estimation in the early fault-tolerant regime, with applications such as overlap certification, trial-state verification, and Monte Carlo methods.
Non-adaptive quantum amplitude estimation (QAE) fixes its Grover depths in advance, so every circuit can run in parallel, but it has so far needed more queries than the best adaptive methods. We show that most of this gap comes from two conventional choices: subspace-based post-processing and power-of-two depth ladders. We replace the first by the exact maximum-likelihood estimate, one matrix multiplication per batch of estimates, and the second by a geometric ladder with ratio r≈1.45r \approx 1.45r≈1.45. The result is a fully parallel, deterministic-schedule estimator with total query complexity 2.82.82.8-3.1/ε3.1/\varepsilon3.1/ε at 95% confidence for target errors from 3.5×10−33.5\times 10^{-3}3.5×10−3 to 10−610^{-6}10−6. This matches the average-case complexity of chebAE, the best benchmarked adaptive method, within statistical uncertainty (with the lower point estimate at every scale tested), beats its maximum-observed complexity by 1.6×1.6\times1.6×, and needs a maximum sequential depth of only 0.21/ε0.21/\varepsilon0.21/ε against chebAE's 2.9/ε2.9/\varepsilon2.9/ε. Relative to csAE, the best non-adaptive benchmark, the constants improve by 30-35% at 95% and 1.51.51.5-1.7×1.7\times1.7× at 99% confidence. The optimal ratio has a simple origin. Doubling is the fastest depth growth at which the data can still tell neighboring candidate values apart, so power-of-two ladders sit at the edge of confusion and must buy reliability with extra shots; a slightly denser ladder checks every scale redundantly. An error-probability analysis reproduces the measured failure rates and locates the optimum. The likelihood formulation extends directly to noise-aware estimation, and uniformly scaling the capped ladder covers the depth-limited regime, realizing the optimal trade-off MNtot≈(0.4M N_{\mathrm{tot}} \approx (0.4MNtot​≈(0.4-0.6)/ε20.6)/\varepsilon^20.6)/ε2 within ∼1.1×\sim 1.1\times∼1.1× of the schedule's Cramér-Rao limit.