Single-Atom Verification of the Noise-Resilient and Fast Characteristics of Universal Nonadiabatic Noncyclic Geometric Quantum Gates

J. W. Zhang, L.-L. Yan, J. C. Li, G. Y. Ding, J. T. Bu, L. Chen, S.-L. Su, F. Zhou, and M. Feng

Phys. Rev. Lett. 127, 030502 – Published 16 July 2021

Abstract

Quantum gates induced by geometric phases are intrinsically robust against noise due to the global properties of their evolution paths. Compared to conventional nonadiabatic geometric quantum computation, the recently proposed nonadiabatic noncyclic geometric quantum computation (NNGQC) works in a faster fashion while still remaining the robust feature of the geometric operations. Here, we experimentally implement the NNGQC in a single trapped ultracold ${}^{40}{Ca}^{+}$ ion to verify the noise-resilient and fast feature. By performing unitary operations under imperfect conditions, we witness the advantages of the NNGQC with measured fidelities by quantum process tomography in comparison to other two quantum gates by conventional nonadiabatic geometric quantum computation and by straightforward dynamical evolution. Our results provide the first evidence confirming the possibility of accelerated quantum information processing with limited systematic errors even in an imperfect situation.

Received 8 March 2021
Accepted 16 June 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.030502

Physics Subject Headings (PhySH)

Adiabatic quantum optimization Quantum gates Quantum information with trapped ions

Trapped ions

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

J. W. Zhang^1,3, L.-L. Yan², J. C. Li^1,3, G. Y. Ding^1,3, J. T. Bu^1,3, L. Chen ¹, S.-L. Su^2,*, F. Zhou^1,†, and M. Feng ^1,2,3,4,‡

¹State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy of Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
²School of Physics, Zhengzhou University, Zhengzhou 450001, China
³School of Physics, University of the Chinese Academy of Sciences, Beijing 100049, China
⁴Research Center for Quantum Precision Measurement, Institute of Industry Technology, Guangzhou and Chinese Academy of Sciences, Guangzhou 511458, China

^*Corresponding author. slsu@zzu.edu.cn
^†Corresponding author. zhoufei@wipm.ac.cn
^‡Corresponding author. mangfeng@wipm.ac.cn

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Vol. 127, Iss. 3 — 16 July 2021

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Images

Figure 1
Single-qubit NNGQC. (a) Level scheme of the trapped ${}^{40}{Ca}^{+}$ ion, where the qubit is encoded in the ground state $| 4^{2} S_{1 / 2}, m_{J} = + 1 / 2 ⟩$ and the metastable state $| 3^{2} D_{5 / 2}, m_{J} = + 1 / 2 ⟩$ . $| 4^{2} P_{1 / 2} ⟩$ is the excited state employed for qubit readout. (b) Conceptual sketch of the evolution trajectories on the Bloch sphere for realizing single-qubit NNGQC gates, where the noncyclic geometric phase is relevant to the solid angle enclosed by the blue solid trajectory and the purple dashed geodesic between A (A’) and B (B’), connecting the initial and final points. The realistic trajectories are determined by the gate-relevant parameters [62] and the geodesic connection is subject to the geodesic rule proposition [64]. The evolution trajectories of $| Ψ_{1} ⟩$ and $| Ψ_{2} ⟩$ are orthogonal and simultaneously reverse. (c1) Pulse sequences to realize the $U_{1}$ gate defined as $U_{1} \equiv [1 + i - 1 - i; 1 - i 1 - i] / 2 = (I + i σ_{z}) / 2 - i (σ_{x} + σ_{y}) / 2$ . (c2),(c3) Experimental results of $U_{1}$ gate reaching the fidelity 0.9796(18) in the evolution started from the initial state $| ψ ⟩ = (| g ⟩ - i | e ⟩) / \sqrt{2}$ , where $P_{e}$ is the population in state $| e ⟩$ and $P_{g e}$ is the off-diagonal term in the evolution matrix. (d) Observed fidelity of $U_{1}$ gate for different input states $| ψ (θ) ⟩ = \cos (θ / 2) | g ⟩ - i \sin (θ / 2) | e ⟩$ with an average fidelity 0.9784(23). The error bars indicating the statistical standard deviation of the experimental data are obtained by 20 000 measurements for each data point.
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Figure 2
Experimental implementation of a single-qubit NNGQC for (a1) a $U_{1}$ gate as defined in Fig. 1 and (a2) a Hadamard gate $H$ , where the input states are eigenvectors of the operator set ${I, σ_{x}, - i σ_{y}, σ_{z}}$ regarding the basis states ${| g ⟩, | e ⟩}$ . (b)–(d) The experimental (black) and numerical (red) bar charts for the real and imaginary parts of the process matrix $χ_{\exp, num}$ , respectively, where the gray parts above the experimental (black) bars denote the errors of the experimental data acquired by 20 000 measurements. The numbers labeled in the axes represent four input states, i.e., $| g ⟩$ , $| e ⟩$ , $| + ⟩ = (| g ⟩ + | e ⟩) / \sqrt{2}$ , and $| - ⟩ = (| g ⟩ - i | e ⟩) / \sqrt{2}$ . (b),(c) A $U_{1}$ gate with the process fidelity of 0.9658(34); (d),(e) a Hadamard gate with the process fidelity of 0.9645(27).
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Figure 3
Comparison of an NNGQC with an NGQC and DQC. (a)–(c) Schematic for processing a $U_{1}$ gate using the three gate schemes with $Ω_{m}$ , the constant Rabi frequency, as explained in [62]. (d),(e) Histogram charts for the real and imaginary parts of $χ$ under different quantum gate processes, which focus on the data around 0 and 0.25 to display the difference among different gate schemes and distinguish small errors of the experimental data. To distinguish the slight differences, we increase the measurement times to 50 000 for each data point. For a full comparison, we also plot the ideal $U_{1}$ gate here, labeled as “IQC.”
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Figure 4
Comparison of the robustness against systematic errors based on the experimental fidelities of the output states after undertaking the $U_{1}$ gates as carried out, respectively, by the NNGQC, NGQC, and DQC schemes for the input state $| g ⟩$ for the Rabi frequency error $δ Ω$ of the laser (a) and for the resonance frequency error $Δ$ of the qubit (b). The experimental results are consistent with the numerical simulation results. The error bars indicating the statistical standard deviation of experimental data are obtained by 20 000 measurements for each data point.
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