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arXiv:2602.17361 · npj Quantum Information

Superiority of Krylov shadow tomography in estimating quantum Fisher information: From bounds to exactness

Yuan-Hao Wang, Da-Jian Zhang

quant-phSubmitted Feb 19, 2026• Announced Feb 20, 2026• 1 citation• Revised Apr 29, 2026 (v2)
v1 · Feb 19, 2026v2 · Apr 29, 2026
arXivHTMLPDF

Abstract

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.

Journal reference
npj Quantum Information 12, 74 (2026)
DOI
10.1038/s41534-026-01216-z
Comments
Published version, 9+4pages, 6 figures

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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.
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.
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.
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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.
Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown ddd-dimensional quantum state ρρρ and a known collection of observables E1,…,EME_1,\ldots,E_ME1​,…,EM​, the goal is to estimate all expectation values {Tr(ρEi)}i=1M\{\text{Tr}(ρE_i)\}_{i=1}^M{Tr(ρEi​)}i=1M​ to additive accuracy ε\varepsilonε with probability at least 1−δ1-δ1−δ. An elusive open question from the seminal shadow tomography work of Aaronson is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on MMM, as suggested by the best-known lower bounds. In this work, we propose two different quantum protocols for shadow tomography with the best sample complexity O(log⁡(M)log⁡(M/δ)ε2), O\left( \frac{\log(M)\log(M/δ)}{\varepsilon^2} \right), O(ε2log(M)log(M/δ)​), which is polylogarithmic in the number of observables and independent of the dimension of the unknown state, thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography from Sinha (STOC 2025) and, more recently, Chen, O'Donnell, Pelecanos, and Wright. Our approach first reduces the general shadow tomography problem to a finite-ensemble estimation problem via a minimax argument. We then develop an observable-independent protocol that repeatedly applies the pretty-good measurement while updating the prior distribution over the finite ensemble according to the measurement outcomes. A tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables and a cubic-logarithmic upper bound. We also introduce a refined recovery-label measurement for the same finite ensemble, which yields the bound in our main theorem.
We establish a direct correspondence between the Krylov hierarchy of quantum Fisher information (QFI) and a Krylov-resolved decomposition of the symmetric logarithmic derivative (SLD). Building on the existing Krylov formulation of QFI, we show that the metrological information resolved at each Krylov level is exactly encoded in the incremental gain of the Lanczos hierarchy, giving the Krylov distribution a direct information-theoretic interpretation. A dual spectral formulation distinguishes the Hilbert--Schmidt geometry used to construct the computational Lanczos hierarchy from the state-dependent geometry in which the SLD and QFI are naturally defined, and relates their associated Liouville measures. This distinction clarifies the origin of different convergence regimes: we recover exponential convergence for gapped spectra and derive an exact algebraic hard-edge convergence law for the Jacobi class, including the exact finite-order error and the marginal case separating finite from divergent QFI.