Recent progress of quantum simulation of non-Hermitian systems

Gao Xue-Er; Li Dai-Li; Liu Zhi-Hang; Zheng Chao

doi:10.7498/aps.71.20221825

Abstract

Quantum simulation is one of the main contents of quantum information science, aiming to simulate and investigate poorly controllable or unobtainable quantum systems by using controllable quantum systems. Quantum simulation can be implemented in quantum computers, quantum simulators, and small quantum devices. Non-Hermitian systems have aroused research interest increasingly in recent two decades. On one hand, non-Hermitian quantum theories can be seen as the complex extensions of the conventional quantum mechanics, and are closely related to open systems and dissipative systems. On the other hand, both quantum systems and classical systems can be constructed as non-Hermitian systems with novel properties, which can be used to improve the precision of precise measurements. However, a non-Hermitian system is more difficult to simulate than a Hermitian system in that the time evolution of it is no longer unitary. In this review, we introduce recent research progress of quantum simulations of non-Hermitian systems. We mainly introduce theoretical researches to simulate typical non-Hermitian quantum systems by using the linear combinations of unitaries, briefly showing the advantages and limitations of each proposal, and we briefly mention other theoretical simulation methods, such as quantum random walk, space embedded and dilation. Moreover, we briefly introduce the experimental quantum simulations of non-Hermitian systems and novel phenomena in nuclear magnetic resonance, quantum optics and photonics, classical systems, etc. The recent progress of the combinations of quantum simulation and non-Hermitian physics has promoted the development of the non-Hermitian theories, experiments and applications, and expand the scope of application of quantum simulations and quantum computers.

Keywords:

Authors and contacts

College of Science, North China Universty of Technology, Beijing 100144, China

Corresponding author: Zheng Chao, czheng@ncut.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12175002, 11705004), the Natural Science Foundation of Beijing, China (Grant No. 1222020), and the Development of Talents Project for Outstanding Young Scholars of Beijing Municipal Institutions, China

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图 1 PT对称和PPH系统的参数空间($ w, s, \theta $ 和$ v, u, \theta $, 设置$ r = 2 $)　(a) PT对称系统; (b) PPH系统^[38]

Figure 1. Parameter spaces of PT-symmetric and P-pseudo-Hermitian systems ($ w, s, \theta $ and $ v, u, \theta $ with setting $ r = 2 $): (a) PT-symmetric systems; (b) PPH systems^[38]

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图 2 APT和APPH系统的参数空间($ w, s, \theta $和$ v, u, \theta $, 设置$ r = 2 $)　(a) APT系统; (b) APPH 系统^[38]

Figure 2. Parameter spaces of APT-symmetric and anti-P-pseudo-Hermitian systems ($ w, s, \theta $ and $ v, u, \theta $ with setting $ r = 2 $): (a) APT-symmetric systems; (b) APPH systems^[38]

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图 3 广义PT对称二能级量子系统的量子线路^[51]

Figure 3. Quantum circuit for a general PT-symmetric two-level system^[51]

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图 4 由辅助qutrit和辅助qudit构造广义PT对称二能级量子系统的电路图　(a)辅助qutrit; (b)辅助qudit^[51]

Figure 4. Quantum circuit for a general PT-symmetric two-level system by ancillary qutrit or ancillary qudit: (a) Ancillary qutrit; (b) ancillary qudit^[51]

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图 5 模拟处于任意相的PT反对称二能级系统的量子线路^[55]

Figure 5. Quantum circuit for a generalized APT-symmetric two-level system in arbitrary phase^[55]

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图 6 量子计算机的流程图和量子线路图　(a)模拟广义APT系统的流程图; (b)模拟广义APT系统的线路图; (c)第一次测量之后的初始化和空间准备的量子线路图; (d)第二次测量之后的量子线路图^[55]

Figure 6. Flow chart and quantum circuit for a qubit computer: (a) Flow chart of quantum simulation of the generalized APT-symmetric system; (b) quantum circuit to simulate the evolution of the generalized APT-symmetric system; (c) quantum circuit for space preparation and initialization after the first measurement; (d) quantum circuit for initialization after the second measurement^[55]

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图 7 Qubit-qudit混合量子线路(由一个工作量子比特和四维辅助量子比特组成的混合系统)^[99]

Figure 7. Qubit-qudit hybrid quantum circuit (The hybrid system consists of a work qubit and a four-dimensional ancillary qudit)^[99]

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图 8 三量子比特线路(由一个工作比特和两个辅助量子比特子系统构成)^[99]

Figure 8. Three-qubit quantum circuit(consists of a work qubit and a two-qubit ancillary subsystems)^[99]

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图 9 (a) qubit-qutrit混合量子计算机的线路图; (b)三比特量子计算机的量子线路图^[101]

Figure 9. (a) Quantum circuit for a qubit-qutrit hybrid computer; (b) quantum circuit designed for a quantum computer of three qubits^[101]

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图 10 (a) qubit-qudit混合量子计算机的线路图; (b)六维子空间中三量子比特量子计算机的线路图^[101]

Figure 10. (a) Quantum circuit for a qubit-qudit hybrid computer; (b) quantum circuit designed for a quantum computer of three qubits using the full Hilbert space^[101]

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图 11 模拟T-APH二能级系统的三量子比特线路图^[57]

Figure 11. Three-qubit quantum circuit to simulate a T-anti-pseudo-Hermitian two-level system^[57]

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图 12 模拟PT-APH二能级系统的三量子比特线路图^[57]

Figure 12. Three-qubit quantum circuit to simulate a general PT-anti-pseudo-Hermitian two-level system^[57]

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图 13 由制备、演化和检测三个模块组成的实验装置^[102]

Figure 13. Experimental configuration includes three modules: the preparation module, the evolution and the detection part^[102]

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图 14 可区分性测量的信息流实验结果^[56]

Figure 14. Experimental results of information flow measured by distinguishability^[56]

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图 15 量子时空和四面体　(a)静态四维(4D) 量子时空; (b)五价点的动态量子时空; (c) S ³ 的局域结构; (d)量子几何四面体^[133]

Figure 15. Quantum spacetime and tetrahedra: (a) A static 4D quantum spacetime; (b) a dynamical quantum spacetime with a number of five valent vertices; (c) the local structure of S ³; (d) quantum geometrical tetrahedra^[133]

DownLoad: Full-Size Img PowerPoint

图 16 实验制备量子态在Bloch 球上的对应和相关经典的四面体^[133]

Figure 16. Experimentally prepared states on the Bloch sphere and their corresponding classical tetrahedra^[133]

DownLoad: Full-Size Img PowerPoint

图 17 LCU模拟YBE的简图^[49]

Figure 17. Schematic illustration of the LCU simulation of the YBE by quantum optics system and a nuclear magnetic resonance quantum system^[49]

DownLoad: Full-Size Img PowerPoint

图 18 用于准备和实现在三模平行高斯光束状态下的算符的实验装置^[152]

Figure 18. Experimental setup used to prepare and to implement the operations on a three-path parallel Gaussian beam state^[152]

DownLoad: Full-Size Img PowerPoint

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