Quantum amplitude estimation can reduce the sampling cost of rare-event probability estimation, but applying it to correlated Ising graphical models is limited by the difficulty of preparing the target distribution and building a practical event oracle. This work explores two approximate strategies for mitigating these challenges. We introduce a sample-free state-preparation method combining loopy belief propagation with the Chow--Liu algorithm. The resulting tree approximation is compiled into a quantum circuit with linear gate count and depth, and its accuracy is evaluated across graph families spanning different topologies, coupling strengths, and coupling signs. We also construct a structural oracle that evaluates threshold rules with reversible Boolean gates. Using a twenty-node supply-chain disruption model as a case study, we compare maximum likelihood amplitude estimation against four classical Monte Carlo baselines. Under the fixed-depth schedule used throughout this work, the quantum estimator has the same asymptotic error scaling as the classical methods but achieves lower estimation error by a constant factor. This reduction narrows when amplitude-encoding queries replace raw shots as the resource metric. We separate statistical error from the deterministic errors caused by approximate state preparation and oracle construction, and identify the requirements for achieving an improvement beyond a constant factor.
Quantum amplitude estimation can reduce the sampling cost of rare-event probability estimation, but applying it to correlated Ising graphical models is limited by the difficulty of preparing the target distribution and building a practical event oracle. This work explores two approximate strategies for mitigating these challenges. We introduce a sample-free state-preparation method combining loopy belief propagation with the Chow--Liu algorithm. The resulting tree approximation is compiled into a quantum circuit with linear gate count and depth, and its accuracy is evaluated across graph families spanning different topologies, coupling strengths, and coupling signs. We also construct a structural oracle that evaluates threshold rules with reversible Boolean gates. Using a twenty-node supply-chain disruption model as a case study, we compare maximum likelihood amplitude estimation against four classical Monte Carlo baselines. Under the fixed-depth schedule used throughout this work, the quantum estimator has the same asymptotic error scaling as the classical methods but achieves lower estimation error by a constant factor. This reduction narrows when amplitude-encoding queries replace raw shots as the resource metric. We separate statistical error from the deterministic errors caused by approximate state preparation and oracle construction, and identify the requirements for achieving an improvement beyond a constant factor.
Structural reliability analysis often requires estimating small failure probabilities under uncertainty, a task for which direct Monte Carlo simulation becomes inefficient because failure observations are scarce. Quantum amplitude estimation offers a potential quadratic improvement in query complexity for bounded expectation estimation, but practical iterative formulations require reliable inference from finite, amplified measurement data. This paper develops a Bayesian sequential formulation of iterative quantum amplitude estimation for rare-event structural failure probability estimation. Structural failure is represented as a binary indicator over a finite stochastic ensemble and encoded through a lookup-table oracle, allowing the failure probability to be treated as a quantum amplitude. Measurement outcomes collected at different Grover depths are assimilated through Bayesian updating over the amplitude angle, yielding posterior estimates, credible intervals, and uncertainty-aware convergence diagnostics. The framework is evaluated on stochastic finite-element benchmark problems, including a one-dimensional bar and an L-bracket with stress concentration. The results show that amplitude amplification converts rare failure events into measurable success probabilities, enabling substantially lower estimation errors than direct Monte Carlo simulation under the same idealized oracle-query budget. The Bayesian formulation achieves point-estimation accuracy comparable to maximum-likelihood IQAE while additionally providing posterior uncertainty quantification, credible intervals, and transparent convergence assessment. The study demonstrates Bayesian IQAE as a statistically interpretable proof-of-concept for quantum-assisted rare-event reliability analysis, while relying on idealized oracle access.
Quantum Amplitude Estimation promises to accelerate Monte Carlo integration by replacing the classical ε∝N−1/2 sampling law with the Heisenberg-limited 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-Ry rotational state preparation, Grover-based amplitude amplification and maximum-likelihood inference and is benchmarked against bmax−1∫0bmaxsin2xdx with bmax=π/5 ,given results emerge. First, the noiseless limit it attains 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≤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≥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.
Causal discovery from observational data underpins applications in finance, climate modeling, and machine learning. Constraint-based causal discovery reduces structure learning to a sequence of conditional independence (CI) tests, where each test decides independence by estimating conditional mutual information I(X;Y∣Z) to additive precision τ and thresholding against it. Classically this requires Θ(1/τ2) samples per test, a cost that dominates in the high-precision regime typical of weak dependencies. We present QKLA (Quantum Kullback--Leibler Amplitude estimation), a quantum algorithm that encodes a clipped log-density ratio as a bounded amplitude and applies amplitude estimation to recover a clipped KL expectation. Given coherent oracle access to the relevant distributions and a reversible log-ratio arithmetic oracle, QKLA achieves a quadratic precision improvement, needing only O((L/τ)log(1/δ)) queries, where L is the log-ratio clip bound. Under per-stratum conditional-oracle access and a margin assumption for CI decisions, embedding this estimator in the PC algorithm compounds to an Ω(1/(Lτ)) reduction in total oracle queries. We validate the theory in three experiments. A gate-level state-vector simulation of the full QKLA circuit confirms the predicted O(1/M) error decay. Across K=20 random binary distributions, classical and quantum error scalings match theory to within 0.01 in slope. In an oracle-model benchmark inside PC on two networks (\textsc{Asia}, 8 nodes; \textsc{Synthetic-12}, 12 nodes), the quantum CI subroutine reaches comparable skeleton-recovery F1 while using 2.7--3.2× fewer oracle queries at τ=5⋅10−3 bits and 4.0--7.4× fewer at τ=10−3 bits.
Quantum state preparation is a fundamental primitive in quantum algorithms for encoding classical data into quantum amplitudes. We compare the cost of preparing general n-qubit states with real amplitudes using two common paradigms: rotation-based methods, based on controlled rotations, and sampling-based methods, based on a structured representation of the target state. Although these approaches are often theoretically compared using CNOT count and T-count, their relative performance in total gate count remains less well understood practically. We compare representative rotation-based and sampling-based methods using T-count and total gate count, and analyze how compilation overhead affects their relative performance. We also develop a software package for compiling state preparation circuits, designed as a practical subroutine for more general quantum computations. Numerical experiments on resource states and quantum states related to quantum chemistry, condensed matter physics, and simulation via Magnus expansion over a range of target accuracies ε support the analysis. Our results show that sampling-based methods achieve asymptotically lower T-count and retain an overall advantage after accounting for total gate count and compilation overhead.
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≈1.45. The result is a fully parallel, deterministic-schedule estimator with total query complexity 2.8-3.1/ε at 95% confidence for target errors from 3.5×10−3 to 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×, and needs a maximum sequential depth of only 0.21/ε against chebAE's 2.9/ε. Relative to csAE, the best non-adaptive benchmark, the constants improve by 30-35% at 95% and 1.5-1.7× 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≈(0.4-0.6)/ε2 within ∼1.1× of the schedule's Cramér-Rao limit.
We establish theoretical foundations for applying quantum amplitude estimation to Wright– Fisher population genetics, demonstrating that quantum methods could provide substantial query complexity improvements — reducing sample requirements from [Formula: see text] to [Formula: see text] — conditional on efficient oracle construction. Through paired classical simulations comparing high-variance vs low-variance estimation strategies across extensive computational replicates, we isolate the effects of variance reduction on evolutionary dynamics while validating theoretical predictions with exceptional precision. Our comprehensive statistical analysis confirms distributional equivalence between methods through consensus across multiple independent tests, including Bayesian model comparison and generational Kolmogorov–Smirnov testing. Overlapping confidence intervals for fixation probability validate that query complexity reduction does not degrade evolutionary prediction quality — the central requirement for practical quantum advantage. Fixation timing distributions remain statistically indistinguishable, demonstrating that variance reduction preserves both stochastic establishment dynamics and deterministic sweep phases across the complete evolutionary trajectory. These theoretical advantages depend entirely on three unresolved prerequisites: (i) oracle construction overhead must remain polynomial in problem parameters, requiring either quantum random access memory (qRAM) — experimentally undemonstrated at practical scales — or problem-specific circuits for fitness landscape encoding; (ii) fault-tolerant quantum hardware with gate error rates orders of magnitude below current technology; (iii) scalable quantum architectures maintaining coherence across the required computational depth. Break-even analysis establishes that without efficient oracle construction, query complexity advantages may be offset by oracle overhead, resulting in marginal or zero net computational benefit. This work contributes: (a) rigorous characterization of variance propagation through evolutionary dynamics with near-perfect scaling law validation, (b) identification of weak-selection, high-noise regimes where quantum variance reduction provides maximum benefit and (c) transparent delineation of necessary vs sufficient conditions for practical quantum advantage. Our framework establishes that reduced estimation variance, if achievable through quantum or other advanced methods, yields measurable evolutionary benefits while explicitly acknowledging that translating theoretical query advantages into end-to-end computational speedup requires coordinated breakthroughs in quantum algorithmic primitives and hardware capabilities.