Metrological Characterization of Non-Gaussian Entangled States of Superconducting Qubits

Kai Xu, Yu-Ran Zhang, Zheng-Hang Sun, Hekang Li, Pengtao Song, Zhongcheng Xiang, Kaixuan Huang, Hao Li, Yun-Hao Shi, Chi-Tong Chen, Xiaohui Song, Dongning Zheng, Franco Nori, H. Wang, and Heng Fan

Phys. Rev. Lett. 128, 150501 – Published 12 April 2022

Abstract

Multipartite entangled states are significant resources for both quantum information processing and quantum metrology. In particular, non-Gaussian entangled states are predicted to achieve a higher sensitivity of precision measurements than Gaussian states. On the basis of metrological sensitivity, the conventional linear Ramsey squeezing parameter (RSP) efficiently characterizes the Gaussian entangled atomic states but fails for much wider classes of highly sensitive non-Gaussian states. These complex non-Gaussian entangled states can be classified by the nonlinear squeezing parameter (NLSP), as a generalization of the RSP with respect to nonlinear observables and identified via the Fisher information. However, the NLSP has never been measured experimentally. Using a 19-qubit programmable superconducting processor, we report the characterization of multiparticle entangled states generated during its nonlinear dynamics. First, selecting ten qubits, we measure the RSP and the NLSP by single-shot readouts of collective spin operators in several different directions. Then, by extracting the Fisher information of the time-evolved state of all 19 qubits, we observe a large metrological gain of ${9.89}_{- 0.29}^{+ 0.28} dB$ over the standard quantum limit, indicating a high level of multiparticle entanglement for quantum-enhanced phase sensitivity. Benefiting from high-fidelity full controls and addressable single-shot readouts, the superconducting processor with interconnected qubits provides an ideal platform for engineering and benchmarking non-Gaussian entangled states that are useful for quantum-enhanced metrology.

Received 29 June 2021
Accepted 15 March 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.150501

Physics Subject Headings (PhySH)

Quantum entanglement Quantum information with solid state qubits Quantum metrology Quantum parameter estimation Quantum simulation

Superconducting qubits

Atomic, Molecular & Optical

Authors & Affiliations

Kai Xu^1,*, Yu-Ran Zhang ^2,3,*, Zheng-Hang Sun^1,*, Hekang Li ¹, Pengtao Song¹, Zhongcheng Xiang¹, Kaixuan Huang¹, Hao Li¹, Yun-Hao Shi ¹, Chi-Tong Chen¹, Xiaohui Song¹, Dongning Zheng¹, Franco Nori ^2,3,4,†, H. Wang ^5,‡, and Heng Fan^1,6,§

¹Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
²Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
³RIKEN Center for Quantum Computing (RQC), Wako-shi, Saitama 351-0198, Japan
⁴Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
⁵Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China
⁶Beijing Academy of Quantum Information Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China

^*These authors contributed equally to this work.
^†fnori@riken.jp
^‡hhwang@zju.edu.cn
^§hfan@iphy.ac.cn

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Vol. 128, Iss. 15 — 15 April 2022

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Images

Figure 1
Superconducting quantum processor and experimental pulse sequence. (a) Simplified schematic of the superconducting quantum processor, showing 19 qubits interconnected by the central bus resonator $R$ . (b),(c) Effective all-to-all coupling strengths $χ_{i j}$ for (b) selected 10 qubits and (c) all 19 qubits. (d) Experimental waveform pulse sequences for detecting (i) the squeezing parameters (RSP and NLSP) and (ii) the FI. Ten or 19 qubits are initially prepared at $| 0 ⟩$ at their idle points and then transformed to $| + ⟩$ by a collective $Y_{π / 2}$ gate. After the free evolution with a time $t$ , when all qubits are equally detuned to the resonator $R$ with $Δ / 2 π ≃ - 580 MHz$ , all qubits at their idle points are measured in the same direction.
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Figure 2
Linear RSP versus NLSP for ten superconducting qubits. (a),(b) The measured matrices $M$ to optimize (a) the linear RSP at $t = 34 ns$ and (b) the NLSP at $t = 50 ns$ , respectively, compared with the numerical simulations. (c) Experimental data of the Husimi $Q$ functions of the states $Q (θ, ϕ)$ against $θ$ and $ϕ$ at specific times with the rotations along the $x$ axis to widen the equatorial distributions. At $t = 0$ , 34, and 82 ns, the $Q (θ, ϕ)$ represent the spin-coherent state, the spin-squeezed state, and the non-Gaussian state, respectively. Insets: experimentally measured $Q (θ, ϕ)$ of the evolved states displayed in spherical polar. (d) Time evolutions of the inverse linear RSP $ξ_{R}^{- 2}$ , the inverse NLSP $ξ_{NL}^{- 2}$ , with a family of operators ${\hat{S}}_{\exp}$ , and the normalized FI $F / N$ compared with the numerical simulations without decoherence (dashed curves). The experimental procedure for extracting the FI from the squared Hellinger distance is shown in Fig. 3. The green solid curve shows the numerical simulation of the normalized quantum FI, $F_{Q} / N = 4 \max_{\hat{n} \in R^{3}} (Δ_{ρ_{t}} {\hat{J}}_{\hat{n}})^{2} / N$ , without decoherence, which is the largest normalized FI over all possible measurements and linear generators. The error bars, indicating the standard deviations of the results, are calculated from 200 000 repetitive experimental runs in total (see Supplemental Material [47] for details).
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Figure 3
Experimental procedure for extracting the FI for 19 superconducting qubits. (a) Schematic of the preparation, nonlinear evolution, optimization rotation, measurement, and comparison of the readouts of two states, with and without the collective phase pulse $Y_{θ}$ , $\exp (- i {\hat{J}}_{y} θ)$ , as in the Ramsey interferometer, for the extraction of the FI. (b) Experimental data of the $Q$ functions $Q (θ, ϕ)$ representing the states after (i) the initial preparation, (ii) the nonlinear evolution with $t = 48 ns$ , (iii) the collective optimization rotation $X_{α}$ , $\exp (- i {\hat{J}}_{x} α)$ , with $α = - 0.288 rad$ , and (iv) the Ramsey pulse $Y_{θ}$ , respectively. (c) The probability distributions $P_{z} (θ)$ of the measurement observable ${\hat{J}}_{z}$ from the single-shot readout of each qubit, for $θ = 0$ , $- 0.05$ , and $- 0.09 rad$ .
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Figure 4
Quantum-enhanced metrology with 19 superconducting qubits. (a) The squared Hellinger distance (HD) at $t = 48 ns$ versus the phase $θ$ in the Ramsey interferometer for different tomography angles $α = - 0.288$ , $- 0.212$ , and $- 0.138 rad$ (along the $x$ axis). The solid lines are for the quadratic curve fitting. (b) The normalized FI $F / N$ extracted from the squared Hellinger distance versus $α$ . The optimal angle at $t = 48 ns$ is obtained as $α_{opt} = - 0.288 rad$ . (c) The time evolution of the normalized FI $F / N$ is compared with the numerical simulations of the inverse RSP (red dashed curve) $ξ_{R}^{- 2}$ and the normalized quantum FI (blue solid curve), $F_{Q} / N = 4 \max_{\hat{n} \in R^{3}} (Δ_{ρ_{t}} {\hat{J}}_{\hat{n}})^{2} / N$ , without decoherence. The error bars, indicating the standard deviations of the results, are calculated from about 600 000 repetitive experimental runs in total (see Supplemental Material [47] for details).
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