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arXiv:2605.14338

Stopping Reliability in Adaptive Krylov-Shadow Quantum Fisher Information Estimation

Erjie Liu, Yangshuai Wang

quant-phSubmitted May 14, 2026• Announced May 15, 2026• 0 citations
v1 · May 14, 2026v2 · May 25, 2026
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Abstract

Adaptive quantum Fisher information (QFI) estimation requires a stopping rule that distinguishes accuracy from apparent numerical stability. For Krylov-shadow QFI estimators, finite Krylov order $K$ produces truncation bias, while finite sample budget $M$ produces finite-$M$ sampling-side error. We show that a width-only empirical stopping rule, based on interval width and local Krylov stability, can declare convergence at small $(K,M)$ even when the post hoc error exceeds the requested tolerance; we call this event a \emph{false stop}. The mechanism is a narrow empirical interval centered on a biased low-$K$ estimate. We give a two-component stopping analysis that separates the Krylov and sampling terms, and we implement a guarded rule that permits a success declaration only after minimum thresholds in $K$ and $M$ and a persistence condition are satisfied. On a five-level dephasing benchmark at $n=4$ qubits, the guarded rule suppresses the false success declarations produced by the width-only empirical rule, whose false-stop rates range from $0.16$ to $0.68$ across the tested noise levels. Under the main fixed resource limit, the guarded rule refuses to make success declarations rather than accepting biased low-$K$ estimates; a separate true-relative-tolerance sampling-budget sequence then shows that, after Krylov and sampling recalibration, the same decision principle can make success declarations without observed false stops. These results show that stopping reliability is a distinct design requirement for adaptive QFI estimation: sampling precision at fixed $K$ does not by itself establish that Krylov truncation bias is controlled.

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Adaptive quantum Fisher information (QFI) estimation requires a stopping rule that distinguishes accuracy from apparent numerical stability. For Krylov-shadow QFI estimators, finite Krylov order KKK produces truncation bias, while finite sample budget MMM produces finite-MMM sampling-side error. We show that a width-only empirical stopping rule, based on interval width and local Krylov stability, can declare convergence at small (K,M)(K,M)(K,M) even when the post hoc error exceeds the requested tolerance; we call this event a \emph{false stop}. The mechanism is a narrow empirical interval centered on a biased low-KKK estimate. We give a two-component stopping analysis that separates the Krylov and sampling terms, and we implement a guarded rule that permits a success declaration only after minimum thresholds in KKK and MMM and a persistence condition are satisfied. On a five-level dephasing benchmark at n=4n=4n=4 qubits, the guarded rule suppresses the false success declarations produced by the width-only empirical rule, whose false-stop rates range from 0.160.160.16 to 0.680.680.68 across the tested noise levels. Under the main fixed resource limit, the guarded rule refuses to make success declarations rather than accepting biased low-KKK estimates; a separate true-relative-tolerance sampling-budget sequence then shows that, after Krylov and sampling recalibration, the same decision principle can make success declarations without observed false stops. These results show that stopping reliability is a distinct design requirement for adaptive QFI estimation: sampling precision at fixed KKK does not by itself establish that Krylov truncation bias is controlled.
Estimating the quantum Fisher information (QFI) is a central task in quantum science and technology, underpinning the benchmarking of measurement devices and the certification of metrological advantages. Owing to the highly nonlinear dependence of the QFI on the quantum state, achieving efficient estimates that reliably converge to the true value, which is a property known as gap closing, has remained a challenge. Here, we address it experimentally by implementing the recently proposed Krylov shadow tomography (KST) protocol on a versatile photonic platform capable of generating Greenberger-Horne-Zeilinger states of up to six qubits with controllable noise. By comparing KST with leading methods under matched resources, we show that it exhibits the predicted gap-closing behavior and yields a substantially lower mean absolute error. Moreover, we demonstrate that the resulting high-accuracy QFI estimates correctly predict the ultimate precision achievable in a phase estimation experiment, as dictated by the quantum Cramér-Rao bound. These results establish KST as a scalable and practical experimental tool for the efficient characterization and utilization of quantum resources in metrological applications.
Estimating the quantum Fisher information (QFI) is a crucial yet challenging task with widespread applications across quantum science and technologies. The recently proposed Krylov shadow tomography (KST) opens a new avenue for this task by introducing a series of Krylov bounds on the QFI. In this work, we address the practical applicability of the KST, unveiling that the Krylov bounds of low orders already enable efficient and accurate estimation of the QFI. We show that the Krylov bounds converge to the QFI exponentially fast with increasing order and can surpass the state-of-the-art polynomial lower bounds known to date. Moreover, we show that certain low-order Krylov bound can already match the QFI exactly for low-rank states prevalent in practical settings. Such exact match is beyond the reach of polynomial lower bounds proposed previously. These theoretical findings, solidified by extensive numerical simulations, demonstrate practical advantages over existing polynomial approaches, holding promise for fully unlocking the effectiveness of QFI-based applications.
Efficiently estimating the quantum Fisher information (QFI) is pivotal in quantum information science but remains an outstanding challenge for large systems due to its high nonlinearity. In this Letter, we tackle this long-standing challenge by integrating the Krylov subspace method-a celebrated tool from applied mathematics-into the framework of shadow tomography. The integrated technique, dubbed Krylov shadow tomography, enables us to formulate a strict hierarchy of nonpolynomial lower bounds on the QFI, among which the highest one matches the QFI exactly. We show that all the bounds can be expressed as expected values of the inverses of Hankel matrices, which are accessible via shadow tomography. Our Krylov shadow tomography therefore opens up a resource-efficient and experimentally feasible avenue to estimate not only nonpolynomial lower bounds but also the QFI itself.
Estimating the quantum Fisher information (QFI) is a central task in quantum science and technology, underpinning the benchmarking of measurement devices and the certification of metrological advantages. Owing to the highly nonlinear dependence of the QFI on the quantum state, achieving efficient estimates that reliably converge to the true value, which is a property known as gap closing, has remained a challenge. Here, we address it experimentally by implementing the recently proposed Krylov shadow tomography (KST) protocol on a versatile photonic platform capable of generating Greenberger-Horne-Zeilinger states of up to six qubits with controllable noise. By comparing KST with leading methods under matched resources, we show that it exhibits the predicted gap-closing behavior and yields a substantially lower mean squared error. Moreover, we demonstrate that the resulting high-accuracy QFI estimates correctly predict the ultimate precision achievable in a phase estimation experiment, as dictated by the quantum Cram\'{e}r-Rao bound. These results establish KST as a scalable and practical experimental tool for the efficient characterization and utilization of quantum resources in metrological applications.
The Quantum Fisher Information (QFI) metric governs a fundamental duality: it quantifies both how precisely a parameter can be estimated (metrology) and how distinguishable two quantum states are (privacy). We exploit this duality to establish a geometry-aware framework for quantum differential privacy (DP) that replaces isotropic depolarizing noise with direction-dependent noise aligned to the QFI eigenstructure of the quantum embedding. We prove six principal theorems: (1) the minimax-optimal mechanism concentrates the noise budget in the dominant QFI eigenmode, achieving ε=(Δ2/2)λmax⁡(1−cγ)\varepsilon = (Δ^2/2)λ_{\max}(1-cγ)ε=(Δ2/2)λmax​(1−cγ) with O(d/λmax⁡)O(d/λ_{\max})O(d/λmax​) advantage; (2) mixed-state QFI decomposition reveals that dephasing in the adversary's basis increases\textit{increases}increases accessible information, while misaligned-basis dephasing provides constructive privacy amplification from hardware noise; (3) a tight privacy −-− utility uncertainty relation ε⋅(1−F)≥Δ22Tr⁡(F)d\varepsilon \cdot (1 - F) \ge \frac{Δ^2}{2}\frac{\operatorname{Tr}(F)}{d}ε⋅(1−F)≥2Δ2​dTr(F)​; (4) adaptive QFI estimation converging at O(1/n)O(1/\sqrt{n})O(1/n​) yields 1.92×1.92\times1.92× tighter bounds; (5) QFI-aligned composition saturates at O(1)O(1)O(1) versus O(k)O(k)O(k) for standard composition; and (6) hardware noise can be harnessed for privacy amplification. Adversarial vulnerabilities, Wasserstein guarantees, subspace projection, and a zero-knowledge audit protocol follow as corollaries. Results are validated on Qiskit Aer GPU simulations, IBM Quantum hardware (ibm_fez, 156 qubits), and against classical DP baselines, achieving equivalent utility at ε≈0.001\varepsilon \approx 0.001ε≈0.001 versus ε≈4800\varepsilon \approx 4800ε≈4800 for classical DP.
Quantum fidelity estimation is a fundamental task in quantum information and computing to benchmark the quality of quantum states, and hence of quantum processes, on noisy quantum hardware. Point estimates of fidelity by existing methods generally lack the rigorous uncertainty quantification required for reliable preparation and verification of these tasks. While Bayesian and bootstrap frameworks are widely used for uncertainty quantification, they are sensitive to model misspecification or lack formal coverage guarantees in non-asymptotic regimes. This paper introduces a hybrid "Bayesian-informed" conformal prediction-based framework to provide distribution-free, rigorous prediction intervals for quantum state fidelity estimation which can be used, with care, to approximate confidence intervals with respect to the quantum randomness of Born's rule. The framework requires a point estimator, and we consider both classical point estimators as well as data-driven neural net estimators. We evaluate the robustness of the methodology across a diverse range of experimental challenges, including finite-sample noise, tomographic incompleteness, physical model error, and out-of-distribution data. Our results demonstrate that conformalized estimators maintain statistical validity where traditional methods fail, offering a scalable and reliable approach for the verification of modern quantum systems. The best performing variant of our methodology is based on a class-conditional conformal prediction, as this most closely approximates the conditional coverage desired for a true confidence interval.