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Robust Estimation of the Quantum Fisher Information on a Quantum Processor

Vittorio Vitale, Aniket Rath, Petar Jurcevic, Andreas Elben, Cyril Branciard, Benoît Vermersch

Published Aug 21, 2024• 29 citations
Publisher

Abstract

We present the experimental measurement, on a quantum processor, of a series of polynomial lower bounds that to the quantum Fisher information (QFI), a fundamental quantity for certifying multipartite entanglement that is useful for metrological applications. We combine advanced methods of the randomized measurement toolbox to obtain estimators that are robust regarding drifting errors caused uniquely during the randomized measurement protocol. We estimate the QFI for Greenberger-Horne-Zeilinger states, observing genuine multipartite entanglement. Then we prepare the ground state of the transverse-field Ising model at the critical point using a variational circuit. We estimate its QFI and investigate the interplay between state optimization and noise induced by our increasing the circuit depth. Published by the American Physical Society 2024

DOI
10.1103/prxquantum.5.030338

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We present the experimental measurement, on a quantum processor, of a series of polynomial lower bounds that to the quantum Fisher information (QFI), a fundamental quantity for certifying multipartite entanglement that is useful for metrological applications. We combine advanced methods of the randomized measurement toolbox to obtain estimators that are robust regarding drifting errors caused uniquely during the randomized measurement protocol. We estimate the QFI for Greenberger-Horne-Zeilinger states, observing genuine multipartite entanglement. Then we prepare the ground state of the transverse-field Ising model at the critical point using a variational circuit. We estimate its QFI and investigate the interplay between state optimization and noise induced by our increasing the circuit depth. Published by the American Physical Society 2024
We present the experimental measurement, on a quantum processor, of a series of polynomial lower bounds that converge to the quantum Fisher information (QFI), a fundamental quantity for certifying multipartite entanglement that is useful for metrological applications. We combine advanced methods of the randomized measurement toolbox to obtain estimators that are robust against drifting errors caused uniquely during the randomized measurement protocol. We estimate the QFI for Greenberg-Horne-Zeilinger states, observing genuine multipartite entanglement. Then, we prepare the ground state of the transverse field Ising model at the critical point using a variational circuit. We estimate its QFI and investigate the interplay between state optimization and noise induced by increasing the circuit depth.
The quantum Fisher information (QFI) represents a fundamental concept in quantum physics. It quantifies the metrological potential of quantum states in quantum parameter estimation measurements, and is intrinsically related to quantum geometry and multipartite entanglement of many-body systems. Using a nitrogen-vacancy center spin in diamond, we experimentally demonstrate a randomized-measurement method to extract the QFI of the qubit, for both pure and mixed states. We then apply this scheme to a 4-qubit state, using a superconducting quantum computer, and show that it provides access to the sub-QFI, which sets a lower bound on the QFI for general mixed states. We numerically study the scaling of statistical error, considering NNN-qubit states, to illustrate the advantage of our randomized-measurement approach in estimating the QFI and multipartite entanglement. Our results highlight the general applicability of our method to different quantum platforms, including solid-state spin systems, superconducting quantum computers, and trapped ions.
The quantum Fisher information (QFI) is a fundamental quantity of interest in many areas from quantum metrology to quantum information theory. It can in particular be used as a witness to establish the degree of multiparticle entanglement in quantum many-body systems. In this work, we use polynomials of the density matrix to construct monotonically increasing lower bounds that converge to the QFI. Using randomized measurements we propose a protocol to accurately estimate these lower bounds in state-of-the-art quantum technological platforms. We estimate the number of measurements needed to achieve a given accuracy and confidence level in the bounds, and present two examples of applications of the method in quantum systems made of coupled qubits and collective spins.
There have been recently significant progress in quantum technologies to realize physically diverse quantum computation and simulation platforms.The aim of this thesis is to develop theoretical approaches based on measurement protocols to experimentally characterize entanglement generated in these state-of-art quantum technological platforms. In the past few years, protocols belonging to the "randomized measurements toolbox" (RM toolbox) have shown to have the ability to measure bipartite entanglement (entanglement between two parts of a quantum system). These protocols have been implemented in several experimental platforms consisting of quantum simulators and quantum computers beyond the regime of full state tomography.The aim of this manuscript is principally dedicated to the development of new approaches based on the RM toolbox to provide experimental groups with advanced tools and methods to probe new fundamental quantities of interest related to entanglement prepared in large quantum platforms.The first objective of this thesis is to provide a new protocol to optimize the existing RM protocol by reducing the statistical errors in the estimation of the quantities. This is done with two approaches: (i) via importance sampling of local random unitaries and (ii) by implementing common randomized measurements. These techniques automatically enable us to implement the RM toolbox to estimate properties associated to entanglement on larger experimental platforms. Then, a central part of this work proposes an alternate effective protocol to measure a fundamental quantity in quantum metrology known as the "quantum Fisher information" that enables us to characterize the multipartite entanglement content of the underlying quantum state. This novel protocol is based on estimating the QFI via the RM toolbox, as a converging series of lower bound polynomials of the density matrix.Additionally, the manuscript also focuses on solving a practical bottleneck of the current RM toolbox caused by the over-burdening classical cost to post-process experimental RM data to evaluate unbiased estimations of relevant quantities.This motivates the introduction of the "batch shadow formalism" that circumvents this problem and provides an efficient data treatment technique with rigorous performance guarantees to get estimates of quantities accessible from the experimental RM data. Lastly, owing to these developments of the RM toolbox, the manuscript reports estimation of some quantities that were accessed for the first time by experimental implementation of this toolbox on two different architectures of quantum platform consisting of superconducting qubits and trapped ions.
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 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.