Experimental estimation of the quantum Fisher information from randomized measurements

Open Access

Experimental estimation of the quantum Fisher information from randomized measurements

Min Yu, Dongxiao Li, Jingcheng Wang, Yaoming Chu, Pengcheng Yang, Musang Gong, Nathan Goldman, and Jianming Cai

Phys. Rev. Research 3, 043122 – Published 16 November 2021

Abstract

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 $N$ -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.

Received 2 April 2021
Accepted 2 November 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.043122

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum metrology Quantum sensing

Quantum Information, Science & Technology

Authors & Affiliations

Min Yu ^1,*, Dongxiao Li ^1,*,**, Jingcheng Wang¹, Yaoming Chu ¹, Pengcheng Yang¹, Musang Gong¹, Nathan Goldman^2,†, and Jianming Cai ^1,3,4,‡

¹School of Physics, International Joint Laboratory on Quantum Sensing and Quantum Metrology, Huazhong University of Science and Technology, Wuhan 430074, China
²Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium
³Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, China
⁴State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China

^*These authors contributed equally to this work.
^**lidongxiao414@hust.edu.cn
^†ngoldman@ulb.ac.be
^‡jianmingcai@hust.edu.cn

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Vol. 3, Iss. 4 — November - December 2021

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Figure 1
(a) The pulse sequence for measuring the sub-QFI (which is equivalent to the QFI in the present scenario) in a Ramsey experiment, using an NV-center spin quantum sensor in diamond. The NV-center spin is initially polarized in the state $| 0 〉$ by applying a green laser pulse and the $ϕ$ -dependent resource state $| ψ_{ϕ} (0) 〉$ is prepared via a microwave pulse ${\hat{Y}}_{ϕ}$ (i.e., a rotation around the $\hat{y}$ axis by an angle $ϕ$ ). The free evolution of duration $t$ results in the parameter-dependent quantum state $ρ_{θ} = ρ (t)$ with $θ = Δ t$ ; see Eq. (4). The random measurement is implemented by three microwave pulses (blue), with random parameters, followed by spin-dependent fluorescence measurement. (b) The fidelity between the evolved state $ρ (t)$ and the initial state $| ψ_{ϕ} (0) 〉$ obtained from randomized measurements (circle) is compared with standard deterministic projective measurement (diamond). (c) The modified Bures distance $D_{G}^{2}$ between the states $ρ_{θ_{0}}$ and $ρ_{θ}$ as a function of $d θ = θ - θ_{0} = Δ d t$ with $θ_{0} = 3 π / 2$ . The dashed curve is the polynomial fit to the experiment data (circles), while the solid curve represents the quadratic fit, the coefficient of which provides an estimation for the value of the QFI [see Eq. (3)]. The experiment parameters in (b) and (c) include the detuning $Δ = (2 π) 1.459$ MHz, the angle $ϕ = π / 2$ , and the number of random measurements $n = 400$ . The coherence time of the NV-center spin is estimated to be $T_{2}^{*} = 2.58 \pm 0.2 μ s$ . (d) The QFI $F_{θ}$ and the purity $P = Tr [ρ^{2} (t)]$ of the evolved state as a function of the free evolution time $t$ . The noise causes decoherence and the system evolves into mixed states, which is evidenced by the decrease of the QFI. (e) The QFI $F_{θ}$ and the coherence $C = | ρ_{12} (t) |$ of the states after a free evolution time $t = 3 π / (2 Δ)$ from different initial resource states $| ψ_{ϕ} (0) 〉 = cos (ϕ / 2) | 0 〉 + sin (ϕ / 2) | - 1 〉$ . The detuning is $Δ = (2 π) 1.459$ MHz.
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Figure 2
(a) Quantum circuit for the measurement of the sub-QFI for a four-qubit entangled state. The Hadamard gate and the control-NOT gates are used to prepare the GHZ state, which is used for the estimation of the parameter $θ$ as induced by the rotation $R_{z} (θ)$ . Local random matrices $u_{j}$ are applied to realize randomized measurements. (b) The modified Bures distance $D_{G}^{2}$ between the state $ρ_{θ_{0}}$ and $ρ_{θ}$ as a function of $d θ = θ - θ_{0}$ ; here we set $θ_{0} = 0$ . The dashed curve is the polynomial fit to the experiment data (circles) generated by the IBM Quantum Falcon Processor ( $ibmq_belem$ v1.0.3), while the solid curve represents the quadratic fit. The experimental sub-QFI $F_{G} = 4.301 \pm 0.22$ is compatible with the value $F_{G} = 4.4807$ obtained from quantum state tomography (see Appendix pp2). This result provides a lower bound for the exact QFI, which is $F_{θ} = 6.5147$ . The number of randomized measurements is $n = 400$ .
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Figure 3
(a) The number of required measurements $n$ in order to achieve an average relative error $\leq ε$ of the QFI (circles, $γ = 0$ ) and the sub-QFI (triangles, $γ / g = 0.01$ ) as functions of the number of the ions. The dashed line and the dash-dotted line are the corresponding exponential fits $2^{a + b N}$ with $a = 10.63 \pm 0.93, b = 0.21 \pm 0.14$ and $a = 10.89 \pm 0.82, b = 0.23 \pm 0.12$ . The parameters are chosen as $α = 1.5$ , $K = 20$ , $d θ = 1.6 / N$ , and $g = Ω = δ = 1 / T$ , $ε = 9 %$ . (b) The QFI and the QFI density as functions of the evolution time for an 8-ion system (solid line for $γ = 0$ and dashed line for $γ / g = 0.01$ ) during the generation of GHZ state. The dash-dotted line shows the lower bound of the QFI and the QFI density as $γ / g = 0.01$ . The circles and the triangles are the results obtained by random measurements. The number of random measurement is $n = 1000$ ; the other parameters are the same as in (a): $K = 20$ , $α = 1.5$ , and $g = Ω = δ = 1 / T$ .
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Figure 4
The QFI and the QFI density obtained from the collective observable ${\hat{s}}_{0} = \sum_{j = 1}^{8} {| 0 〉}_{j j} 〈 0 | / 8$ [see Eq. (A19)] as functions of the evolution time for the same 8-ion system as that of Fig. 3 in the main text. The relevant parameters are chosen as $γ = 0$ , $d θ = 0.1$ , $n = 2000$ , $K = 20$ , $α = 1.5$ , and $g = Ω = δ = 1 / T$ .
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Figure 5
The modified Bures distance $D_{G}^{2} (ρ_{θ_{0}}, ρ_{θ})$ between the states $ρ_{θ_{0}}$ and $ρ_{θ}$ as a function of $d θ = θ - θ_{0} = Δ d t$ for $θ_{0} = 5 π / 2$ (circles), $9 π / 2$ (squares), and $11 π / 2$ (triangles). The solid, dashed, dash-dotted curves are the quadratic fit of the experiment data, the corresponding coefficients of which provide the values of the QFI as $0.94 \pm 0.13$ , $0.69 \pm 0.10$ , and $0.56 \pm 0.10$ . The other relevant experimental parameters are the detuning $Δ = (2 π) 1.459$ MHz, the angle $ϕ = π / 2$ , and the number of random measurements is $n = 400$ . The coherence time of the NV-center spin is estimated to be $T_{2}^{*} = 2.58 \pm 0.2 μ s$ . We note that the experimental operations are the same as those of Fig. 1 in the main text.
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Figure 6
Density matrix elements of the prepared four-qubit state $ρ$ obtained via Qiskit using the IBM Quantum Falcon Processor ( $ibmq_belem$ v1.0.3). The fidelity with respect to the GHZ state is estimated to be $78 %$ .
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