Experimental demonstration of a digital quantum simulation of a general $\mathcal{PT}$-symmetric system

Experimental demonstration of a digital quantum simulation of a general $PT$ -symmetric system

Jingwei Wen, Chao Zheng, Xiangyu Kong, Shijie Wei, Tao Xin, and Guilu Long

Phys. Rev. A 99, 062122 – Published 28 June 2019

Abstract

$PT$ -symmetric systems are one of the most interesting research fields in modern quantum physics, where various theoretical and experimental progress has been made. In our work, we experimentally demonstrate how to realize a general $PT$ -symmetric two-level operation in a quantum computation frame with a universal circuit model. It is based on enlarging the system with ancillary qubits and encoding the subsystem with the non-Hermitian Hamiltonian with postselection. Furthermore, we use the general formula to demonstrate one particular interesting issue, entanglement restoration induced by local $PT$ -symmetric operation in our four-qubit liquid nuclear magnetic resonance platform, and a theoretical explanation for the physical characteristics has been proposed. We also extend the protocol to the realization of an arbitrary two-level Hamiltonian evolution without a Hermitian restriction by an appropriate modification of the original quantum circuit. Our work lays the foundation for quantum simulation of the general $PT$ -symmetric problem in realistic quantum simulators, and the scheme can be extended to other quantum computation platforms.

Received 25 February 2019

DOI:https://doi.org/10.1103/PhysRevA.99.062122

Physics Subject Headings (PhySH)

Quantum algorithms Quantum entanglement Quantum simulation

Quantum Information, Science & Technology

Authors & Affiliations

Jingwei Wen ¹, Chao Zheng², Xiangyu Kong¹, Shijie Wei^3,*, Tao Xin^4,5,6,†, and Guilu Long^1,7,8,‡

¹State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
²Department of Physics, College of Science, North China University of Technology, Beijing 100144, China
³IBM Research, Beijing 100094, China
⁴Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
⁵Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518055, China
⁶Shenzhen Key Laboratory of Quantum Science and Engineering Southern University of Science and Technology, Shenzhen 518055, China
⁷Tsinghua National Laboratory for Information Science and Technology, Beijing 100084, China
⁸Collaborative Innovation Center of Quantum Matter, Beijing 100084, China

^*liangqq1989626@163.com
^†xint@sustc.edu.cn
^‡gllong@tsinghua.edu.cn

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Vol. 99, Iss. 6 — June 2019

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Figure 1
Quantum circuit for a general $PT$ -symmetric two-level operator. (a) Two-qubit protocol. Single-qubit operators $V$ and $W$ (Hadamard gate) are operated on the ancillary qubit and two two-qubit operators (0-controlled $V_{1}$ and 1-controlled $V_{2}$ ) are implemented in the system, so we achieve $PT$ -symmetric evolution $U_{PT} = e^{- i H_{PT} t / ℏ}$ on the second qubit by collapsing the ancillary qubit into $| 0 〉$ . (b) Three-qubit protocol. A three-qubit quantum circuit for the realization of a $PT$ -symmetric two-level operator. In the circuit, the identity matrix $I$ and the Pauli matrix $X, Y, Z$ form a set of complete basis in two-dimensional Hilbert space.
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Figure 2
(a) Molecule structure of ${}^{13}C$ -labeled crotonic acid. C1, C2, C3, and C4 are used as four qubits in the experiment, while all ${}^{1}H$ are decoupled throughout the experiment. (b) Molecule parameters of the sample: The chemical shifts and $J$ couplings (in Hz) are listed by the diagonal and off-diagonal elements, respectively. T2 (in seconds) are also shown at the bottom. (c) Quantum circuit for demonstrating $PT$ -symmetric Hamiltonian evolution by varying the initial state.
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Figure 3
Experimental results when a $PT$ -symmetric system is observable. We vary $θ$ from 0 to $0.5 π$ (a)–(f) in intervals of $0.1 π$ and plot the real (left in each subfigure) and imaginary (right in each subfigure) part of the final $4 \times 4$ density matrix. The internal solid bars represent the experimental results, while the external transparent bars represent the corresponding theoretical values of the density matrices.
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Figure 4
(a) Quantum circuit for entanglement restoration. Two-qubit operator $V$ is set by parameters in a $PT$ -symmetric Hamiltonian and is related to the evolution time as well. Experimental results are read out by QST and we focus on the subspace when the first two qubits are $| 00 〉$ . (b) Theoretical values of concurrence evolving with time (unit in seconds) when $PT$ symmetry is observable and unobservable are represented with green and blue lines, separately. The experimental results in discrete time are plotted with red stars when $PT$ symmetry is observable and red circles represent experimental results for unobservable $PT$ symmetry.
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Figure 5
Quantum circuit for the realization of a two-qubit operator $V$ and the definitions of the operators are shown in Eq. (A1).
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Figure 6
Quantum gate decomposition for the two-qubit protocol. $R_{y} (2 θ)$ is introduced to change the initial state and we vary $θ$ from 0 to $π / 2$ . The form of the other gates is shown in Table 2.
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Physical Review A

covering atomic, molecular, and optical physics and quantum information

Experimental demonstration of a digital quantum simulation of a general $PT$ -symmetric system

Jingwei Wen, Chao Zheng, Xiangyu Kong, Shijie Wei, Tao Xin, and Guilu Long

Phys. Rev. A 99, 062122 – Published 28 June 2019

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Physical Review A

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Experimental demonstration of a digital quantum simulation of a general PT-symmetric system

Jingwei Wen, Chao Zheng, Xiangyu Kong, Shijie Wei, Tao Xin, and Guilu Long

Phys. Rev. A 99, 062122 – Published 28 June 2019

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Experimental demonstration of a digital quantum simulation of a general $PT$ -symmetric system