Experimental Realization of Nonadiabatic Holonomic Single-Qubit Quantum Gates with Optimal Control in a Trapped Ion

Ming-Zhong Ai, Sai Li, Zhibo Hou, Ran He, Zhong-Hua Qian, Zheng-Yuan Xue, Jin-Ming Cui, Yun-Feng Huang, Chuan-Feng Li, and Guang-Can Guo

Phys. Rev. Applied 14, 054062 – Published 24 November 2020

Abstract

Quantum computation with quantum gates induced by geometric phases is regarded as a promising strategy in fault-tolerant quantum computation, owing to its robustness against operational noise. However, because of the parametric restrictions in previous schemes, the main robust advantage of holonomic quantum gates is reduced. Here, we experimentally demonstrate a solution scheme, obtaining nonadiabatic holonomic single-qubit quantum gates with optimal control in a trapped $^{171} {Yb}^{+}$ ion based on a three-level system with resonant driving, which also has the advantages of rapid evolution and convenient implementation. Compared with previous geometric gates and conventional dynamical gates, the superiority of our scheme is that it is more robust against control amplitude errors, which is confirmed by the gate infidelity as measured by both quantum-process tomography and random benchmarking methods. In addition, we outline how nontrivial two-qubit holonomic gates can also be realized using currently available experimental technology. Thus, our experiment confirms the feasibility of this robust and fast holonomic quantum-computation strategy.

Received 23 June 2020
Revised 9 September 2020
Accepted 2 November 2020

DOI:https://doi.org/10.1103/PhysRevApplied.14.054062

Physics Subject Headings (PhySH)

Adiabatic quantum optimization Geometric & topological phases Quantum control Quantum gates Quantum information with trapped ions

Microwave techniques Resonance techniques

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Ming-Zhong Ai^1,2, Sai Li³, Zhibo Hou^1,2, Ran He^1,2, Zhong-Hua Qian^1,2, Zheng-Yuan Xue ^3,4,*, Jin-Ming Cui^1,2,†, Yun-Feng Huang^1,2,‡, Chuan-Feng Li ^1,2,§, and Guang-Can Guo^1,2

¹CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
²CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
³Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, and School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
⁴Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China

^*zyxue83@163.com
^†jmcui@ustc.edu.cn
^‡hyf@ustc.edu.cn
^§cfli@ustc.edu.cn

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Vol. 14, Iss. 5 — November 2020

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Figure 1
Realization of arbitrary single-qubit nonadiabatic holonomic gates. (a) Hyperfine energy levels of a $^{171} {Yb}^{+}$ ion. The qubit states $| 0 ⟩$ and $| 1 ⟩$ are encoded in $|^{2} S_{1 / 2}, F = 1, m_{F} = 0 ⟩$ , and $|^{2} S_{1 / 2}, F = 1, m_{F} = 1 ⟩$ , respectively. Two microwave fields $ω_{0 a}$ and $ω_{1 a}$ are resonantly coupled with transitions $| 0 ⟩ \leftrightarrow | a ⟩$ and $| 1 ⟩ \leftrightarrow | a ⟩$ to generate holonomic gates in the qubit subspace. (b) Illustration of the implemented holonomic gates in the Bloch sphere of the ${| b ⟩, | a ⟩}$ subspace. The whole evolution is divided into two steps: first evolving from the bright state $| b ⟩$ to the auxiliary state $| a ⟩$ and then back with an additional phase. (c) Our simplified experimental setup. An ion is trapped in a needle trap that has a pair of rf electrodes and two pairs of dc electrodes. The microwaves generated from an AWG mix with microwave from a signal generator (E8257D), and then this mixed signal interacts with the ion through a microwave horn.
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Figure 2
QPT results for holonomic single-qubit quantum gates. (a) Experimental sequences for the QPT. A group of complete bases are used to prepare the initial state, and each final state is reconstructed by performing quantum-state tomography. An arbitrary holonomic single-qubit gate to be characterized is implemented in the middle of these sequences. (b) Bar charts of the real and imaginary parts of the process matrix for the $X$ , $H$ , $T$ , and $S$ gates. The solid black outlines are plotted from the theoretical estimation.
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Figure 3
RB results for holonomic single-qubit quantum gates. (a) Sequences of reference and interleaved RB experiments. (b) Sequence fidelity with the number of Clifford gates. The fidelity for each sequence length is measured for 20 different random sequences, with the standard deviation from the mean plotted as the error bars. All curves are fitted through $F = A p^{m} + B$ . The average gate fidelity is calculated using $F_{ave} = 1 - (1 - p_{ref}) / 2$ , and a specific gate fidelity is calculated using $F_{gate} = 1 - (1 - p_{gate} / p_{ref}) / 2$ , where $p_{ref}$ and $p_{gate}$ are the $p$ parameters for the reference and interleaved RB processes, respectively.
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Figure 4
Noise-resilient features of different types of quantum gates. In (a),(b), the $X$ and $H$ gate infidelities, respectively, are plotted as functions of the Rabi-frequency error for RNHQC with $η = 1$ and conventional NHQC with $η = 0$ . These results indicate the superiority of RNHQC against Rabi-frequency error, especially when the error is large. In (c),(d), the infidelities of both holonomic and dynamical gates are plotted as functions of the Rabi-frequency error for $X$ and $H$ gates, respectively. These results indicate the superior robustness of holonomic gates compared with dynamical ones. The error bars indicate the standard deviation, and each data point is averaged over 2000 realizations.
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Figure 5
Amplitudes and phases of the microwave field used in our experiments. (a),(c) The normalized amplitudes and phases, respectively, of conventional NHQC and RNHQC $X$ gates. (b),(d) The normalized amplitudes and phases, respectively, of RNHQC and dynamical $H$ gates.
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Figure 6
Energy-level diagram of a trapped $^{171} {Yb}^{+}$ ion. Two 355-nm pulsed laser beams perpendicular to each other are used to excite the motional mode. The difference in frequency between these two lasers is $ω_{r 1} - ω_{r 2} = ω_{0 a} + ω_{x}$ , which corresponds to a blue sideband operation. By optimizing the waveform of this blue sideband, a nontrivial two-qubit phase gate can be realized as mentioned in the main text.
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