quantum amplitude estimation

This work investigates the fundamental limits of quantum speedup via amplitude amplification and estimation when efficient inversion of the state-preparation unitary is unavailable—a common scenario in quantum learning, metrology, and sensing where time reversal is infeasible. Method: The authors establish a “reversibility criterion,” proving that efficient invertibility of the state-preparation unitary is necessary for achieving quantum advantage beyond classical sampling complexity. They rigorously refute the possibility of super-classical speedup using only forward unitary operations, via a compressed oracle model and explicit counterexamples based on trace estimation. Contribution/Results: The work provides the first rigorous characterization of reversibility as a foundational prerequisite for amplitude-based quantum acceleration. It demonstrates that Grover-type quadratic speedup generally fails without efficient inversion, thereby explaining the uneven distribution of quantum advantage across domains. These results fundamentally constrain the applicability of amplitude amplification in realistic, non-reversible quantum settings.

Addressing the unreliability of quantum measurements under depolarizing noise—dominant in the NISQ era—this paper proposes a statistically driven quantum error mitigation (QEM) method to accurately estimate the most probable noise-free output from noisy measurement samples. The method introduces an innovative two-stage “filtering + EM” framework: first, a heuristic filtering stage explicitly isolates and suppresses non-informative depolarizing noise; second, expectation-maximization (EM) is applied to the denoised data to enhance both interpretability and scalability of maximum-likelihood estimation. Small-scale experiments using Qiskit demonstrate that the approach significantly outperforms existing statistical QEM techniques. Further validation on synthetic datasets confirms its scalability to systems with ~100 qubits. By unifying theoretical rigor with engineering practicality, this work establishes a novel paradigm for error mitigation in intermediate-scale noisy quantum computation.

This work addresses the challenge of error propagation and unreliable outcomes in noisy quantum computing, arising from both hardware noise and intrinsic stochasticity. It introduces, for the first time, a systematic uncertainty quantification (UQ) framework into quantum computation by formulating the problem as a statistical inference task. By integrating tools from probabilistic modeling, Bayesian inference, stochastic analysis, and sensitivity analysis, the study establishes a novel paradigm for error characterization and algorithm design tailored to noisy intermediate-scale quantum (NISQ) devices. The proposed uncertainty-aware framework is not only scalable but also provides a unified and mathematically rigorous foundation for error verification, characterization, and mitigation strategies.

This work addresses the lack of systematic tools for proving quantum query lower bounds. We establish the first quantum sample-to-query lifting theorem, which rigorously connects sampling complexity to query complexity from an information-theoretic perspective—introducing a new paradigm for quantum lower-bound analysis. The theorem unifies analyses across several fundamental problems: quantum property testing, Gibbs sampling, entanglement entropy estimation, and matrix spectral testing. Specifically, it yields the optimal quadratic separation for quantum state discrimination; proves the query-optimality of Gilyén et al.’s Gibbs sampler; provides a tight (widetilde{Omega}(1/sqrt{Delta})) lower bound on entanglement entropy estimation under (Delta)-spectral gap; and strengthens existing lower bounds for multiple spectral testing tasks. The result combines deep theoretical insight with broad methodological applicability, offering a unified framework for deriving quantum query lower bounds across diverse computational settings.

This work addresses precision enhancement in single-parameter quantum state estimation under the influence of unknown nuisance parameters. To overcome limitations of conventional two-stage measurement schemes—namely, their reliance on restrictive regularity conditions for classical estimators and the stringent applicability requirements of the Quantum Cramér–Rao Bound (QCRB)—we propose a relaxed adaptive two-stage quantum estimation algorithm. In the first stage, a parameter-independent measurement yields an initial estimate; in the second stage, a QCRB-optimal adaptive measurement is performed conditioned on this estimate. This is the first extension of the two-stage framework to quantum sensing scenarios with unknown nuisance parameters, and it significantly broadens the class of admissible classical estimators by relaxing regularity assumptions. We rigorously establish asymptotic √n-consistency and asymptotic normality of the estimator. Furthermore, we derive the exact asymptotic error bound for quantum-enhanced transmission-rate sensing, providing a solid theoretical foundation for practical quantum metrology.

This work addresses the inefficiency of estimating rare failure events in structural reliability analysis by proposing a novel method that integrates quantum amplitude estimation with Bayesian sequential inference. The failure probability is encoded as a quantum amplitude, and amplitude amplification is achieved via Grover iterations employing a lookup-table-based oracle. Leveraging measurement outcomes from varying iteration depths, the method performs sequential Bayesian updating to infer the posterior distribution of the amplitude angle. To the best of our knowledge, this is the first approach to incorporate Bayesian sequential inference into iterative quantum amplitude estimation. Under a fixed oracle query budget, the proposed method significantly outperforms classical Monte Carlo simulation in efficiency, achieves point estimation accuracy comparable to maximum-likelihood-based iterative quantum amplitude estimation, and further provides full uncertainty quantification, credible intervals, and convergence diagnostics—thereby enhancing the statistical interpretability of the results.

This work addresses the query complexity of quantum kernel methods during inference, which typically scales linearly with the number of training samples $N$. By jointly optimizing the strategy for kernel value estimation—choosing between sampling and quantum amplitude estimation—and the summation approach—either estimating terms individually or encoding the entire sum into a single observable—we propose the first inference algorithm with query complexity $O(\|\alpha\|_1/\varepsilon)$, thereby eliminating any dependence on $N$. We prove this complexity is theoretically optimal by establishing a matching lower bound of $\Omega(\|\alpha\|_1/\varepsilon)$, achieving quadratic speedups in both $\|\alpha\|_1$ and $\varepsilon$. Furthermore, we uncover a fundamental trade-off between query-optimal and gate-complexity-optimal strategies, offering practical guidance tailored to different hardware constraints.

This work addresses the challenge of estimating risk-neutral expectations for credit valuation adjustment (CVA) on noisy quantum hardware, where circuit depth limitations and hardware noise degrade accuracy. The authors propose an end-to-end noise-aware quantum workflow that integrates market calibration, discretized modeling, joint spatiotemporal distribution encoding via a quantum circuit Born machine (QCBM), controlled payoff rotation, and error decomposition analysis. Central to this framework is the novel contrast-aware Bayesian iterative quantum amplitude estimation (CABIQAE) algorithm, which embeds experimentally calibrated Grover contrast loss into Bayesian inference and enables adaptive circuit depth selection. Experimental results on real quantum devices demonstrate that CABIQAE significantly outperforms noise-agnostic approaches, incurs substantially lower classical post-processing overhead than existing noise-aware Bayesian amplitude estimation baselines, and provides the first quantitative breakdown of total CVA error contributions from statistical, encoding, discretization, and hardware-induced sources.

This work investigates the sample complexity of estimating the fidelity between an unknown quantum state and a known reference state up to an additive error ε, with a focus on scenarios where either the reference state or the unknown state exhibits low-rank structure. By integrating tools from quantum information theory, statistical estimation, and low-rank assumptions, the authors improve the sample complexity from O(r²log²(1/ε)/ε⁴) to the optimal O(r²/ε²) when the reference state has rank r, and establish a matching lower bound of Ω(r/ε²). The analysis also extends to the setting where the unknown state is low-rank while the reference state is arbitrary. These results yield nearly tight sample complexity bounds across multiple regimes and significantly broaden the theoretical framework for fault-tolerant quantum state certification.

We study the composite sequential quantum hypothesis testing (SQHT) problem, where the objective is to distinguish a null quantum state from a compact, convex set of alternative quantum states. We propose a mixture-sequential quantum probability ratio test that adaptively selects measurements based on the current mixture estimate of the alternative set, and stops upon the first threshold crossing of the mixture log-likelihood ratio. Under an expected sample size constraint, we show that our proposed adaptive strategy simultaneously achieves the optimal Type-I and (worst-case) Type-II error exponents. These exponents are characterized by the minimal measured relative entropies between the null state and the alternative set. We further establish a matching converse, thereby characterizing the optimal error exponent region. Finally, our results show that achieving vanishing error probabilities in composite SQHT requires an expected sample complexity at least as large as that of sequential testing between two fixed quantum states.