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Applied Physics

Quantum Simulation Foundations of Imaginary-Time Evolution of Non-Hermitian Systems

瑞娟 赵

Published Jan 1, 2025• 0 citations
Publisher

Abstract

非厄米量子系统哈密顿量因具有不同的对称性，在量子态快速演化、量子态区分、量子精密测量等方面展现出超越标准量子力学系统的优势和新奇特性，近二十多年来受到持续关注，成为前沿领域。量子模拟非厄米系统是主要方向之一，又是研究非厄米系统的重要手段。典型的非厄米系统包括宇称–时间(Party-Time，简称PT)对称系统和赝厄米对称系统等。已有研究集中于对非厄米系统的时间演化开展量子模拟。文章围绕非厄米系统的虚时演化算符进行研究，通过计算非厄米系统的虚时演化算符，为开展典型非厄米系统的虚时演化量子模拟理论研究打下基础。我们的研究从理论上拓展了量子计算机可模拟新奇系统的范畴，并将虚时演化拓展至非厄米系统。基于我们对非厄米系统的虚时演化理论研究，对利用不同的量子系统和量子真机开展实验研究具有指导作用。The Hamiltonian of non-Hermitian quantum systems, due to its different symmetries, has shown advantages and novel properties beyond standard quantum mechanical systems in terms of rapid quantum state evolution, quantum state differentiation, quantum precision measurement, etc., which has attracted continuous attention and become a frontier field in the past two decades. Quantum simulation of non-Hermitian systems is one of the main directions and an important means to study non-Hermitian systems. Typical non-Hermitian systems include parity-time (PT) symmetric systems and pseudo Hermitian symmetric systems. Previous studies have focused on the quantum simulation of the time evolution of non-Hermitian systems. In this paper, the virtual time evolution operators of non-Hermitian systems are studied. By calculating the virtual time evolution operators of non-Hermitian systems, it lays a foundation for the theoretical study of virtual time evolution of typical non-Hermitian systems. Our research theoretically expands the range of novel systems that quantum computers can simulate, and extends virtual time evolution to non-Hermitian systems. Based on our research on the virtual time evolution theory of non-Hermitian systems, it has a guiding role to carry out experimental research with different quantum systems and quantum real machines.

DOI
10.12677/app.2025.154035

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非厄米量子系统哈密顿量因具有不同的对称性，在量子态快速演化、量子态区分、量子精密测量等方面展现出超越标准量子力学系统的优势和新奇特性，近二十多年来受到持续关注，成为前沿领域。量子模拟非厄米系统是主要方向之一，又是研究非厄米系统的重要手段。典型的非厄米系统包括宇称–时间(Party-Time，简称PT)对称系统和赝厄米对称系统等。已有研究集中于对非厄米系统的时间演化开展量子模拟。文章围绕非厄米系统的虚时演化算符进行研究，通过计算非厄米系统的虚时演化算符，为开展典型非厄米系统的虚时演化量子模拟理论研究打下基础。我们的研究从理论上拓展了量子计算机可模拟新奇系统的范畴，并将虚时演化拓展至非厄米系统。基于我们对非厄米系统的虚时演化理论研究，对利用不同的量子系统和量子真机开展实验研究具有指导作用。The Hamiltonian of non-Hermitian quantum systems, due to its different symmetries, has shown advantages and novel properties beyond standard quantum mechanical systems in terms of rapid quantum state evolution, quantum state differentiation, quantum precision measurement, etc., which has attracted continuous attention and become a frontier field in the past two decades. Quantum simulation of non-Hermitian systems is one of the main directions and an important means to study non-Hermitian systems. Typical non-Hermitian systems include parity-time (PT) symmetric systems and pseudo Hermitian symmetric systems. Previous studies have focused on the quantum simulation of the time evolution of non-Hermitian systems. In this paper, the virtual time evolution operators of non-Hermitian systems are studied. By calculating the virtual time evolution operators of non-Hermitian systems, it lays a foundation for the theoretical study of virtual time evolution of typical non-Hermitian systems. Our research theoretically expands the range of novel systems that quantum computers can simulate, and extends virtual time evolution to non-Hermitian systems. Based on our research on the virtual time evolution theory of non-Hermitian systems, it has a guiding role to carry out experimental research with different quantum systems and quantum real machines.
Since the role that 𝓟𝓣-symmetric Hamiltonians could play in quantum mechanics theory was suggested in1998, much work has been conducted on the mathematical analysis of 𝓟𝓣-symmetry. Carl Benderchallenged the perceived wisdom of Quantum Mechanics that the Hamiltonian operator describing anyquantum mechanical system has to be Hermitian. He showed that Hamiltonians that are invariant undercombined parity-time 𝓟𝓣-symmetry transformations likewise can exhibit real eigenvalue spectra.However, 𝓟𝓣-symmetric systems began as an analytic extension of classical and quantum mechanical tothe complex domain. These findings had a particularly profound impact in the field of photonics, where PTsymmetricpotential landscapes can be implemented by introducing dissipative terms appropriatelydistributing gain and loss in open or coupled systems. The refractive index profile plays the role of thecomplex potential. Similar research activity was transferred to other physical settings such as electronicscircuits, microwaves and cavities resonators, acoustic, superconducting wires, and atomic systems.Unconventional characteristics compared with Hermitian ones were found and allow to engineer newdevices.Likewise, topological phases of matter are another new field of physics which have had a rapid growth inthe last years, based on the concept of Geometric or Berry phase. Topological Insulators andSuperconductors, Topological photonics and nanophotonics, Topological Computers are some of theapplications widely described in scientific journals.In Nature all the processes have to be described by a property in a discrete or iterative way instead of acontinuous way which usually is time. Continuous processes, being a mathematical idealization, has beenthe best tool to model observable properties in physics and is the conventional tool used in Classical andQuantum Mechanics. The formal way to model discrete systems is through difference equations which aresimilar to differential equations but not equal.Identification of dynamical discrete system with recurrence equations, give us the possibility to consider ananalytical continuation of its evolution variable, from discrete values to fractional one´s, in the solution ofthe system. Parametric solutions obtained by this procedure takes automatically complex values, avoidingto make any supposition about the possibility of their analytical extension to complex domain. Thisprocedure leads, in a very simple natural way, a rich variety of dynamical behaviors in the complex domainunknown in the real domain. This new mathematical tool is developed to enhance and provide a moresubtle description of the behavior of dynamical systems and provides a more general class of Non-Hermitian Hamiltonians than 𝓟𝓣-symmetric systems with real spectra, keeping also topological propertiesdue to a kind of geometric phase.
ABSTRACT Non‐Hermitian systems have attracted increasing attention in recent years, particularly those exhibiting Parity‐Time (PT) symmetry. Quantum metrology establishes the ultimate attainable bounds for parameter estimation in both Hermitian and non‐Hermitian systems. However, comparatively little is known about non‐Hermitian systems with anti‐Parity‐Time (APT) symmetry. In this work, we investigate the theoretical bounds on quantum parameter estimation precision for an APT‐symmetric system governed by the Bogoliubov–de Gennes (BdG) Hamiltonian. We explicitly derive the expression for the quantum Fisher information (QFI) and compare the attainable precision limits for both multiplicative and non‐multiplicative parameters in the BdG Hamiltonian. Our analysis of the effective QFI reveals that quantum parameter estimation precision can be significantly enhanced as the system approaches the exceptional point with unbroken APT symmetry, where Heisenberg scaling becomes achievable. We also examine how the effective QFI depends on the choice of initial states. Furthermore, we demonstrate that entanglement in multi‐particle initial state can provide an additional enhancement to the QFI. The ratio of limit of parameter estimation precision of the entangled state to the greatest one we have found over all the tensor product states is 1.22, which is greater than 1. Our analysis of two‐particle states confirms that entanglement in the initial state can further increase the QFI, highlighting it as a valuable quantum resource. These findings lay a theoretical foundation for future experimental explorations of parameter estimation in APT‐symmetric non‐Hermitian systems.
In standard quantum mechanics, the Hamiltonian describing the physical system is generally Hermitian, so as to ensure that the system has real energy spectra and that the system’s evolution is unitary. In recent years, it has been found that non-Hermitian Hamiltonians with parity-time ( \begin{document}PT{\cal {PT}}PT\end{document} ) symmetry also have real energy spectra, and there is a novel non-Hermitian exceptional point between \begin{document}PT{\cal {PT}}PT\end{document} -symmetric phase and \begin{document}${\cal {PT}} $\end{document} -symmetry-broken phase, which is unique to non-Hermitian systems. Recently, people have realized \begin{document}${\cal {PT}} $\end{document} symmetric and anti- \begin{document}${\cal {PT}}$\end{document} symmetric non-Hermitian Hamiltonians in various physical systems and demonstrated novel quantum phenomena, which not only deepened our understanding of the basic laws of quantum physics, but also promoted the breakthrough of application technology. This review will introduce the basic physical principles of \begin{document}${\cal {PT}} $\end{document} symmetry and anti- \begin{document}${\cal {PT}}$\end{document} symmetry, summarize the schemes to realize \begin{document}${\cal {PT}} $\end{document} symmetry and anti- \begin{document}${\cal {PT}} $\end{document} symmetry in optical and atomic systems systematically, including the observation of \begin{document}${\cal {PT}} $\end{document} -symmetry transitions by engineering time-periodic dissipation and coupling in ultracold atoms and single trapped ion, the realization of anti- \begin{document}${\cal {PT}} $\end{document} symmetry in dissipative optical system by indirect coupling, and realizing anti- \begin{document}${\cal {PT}} $\end{document} -symmetry through fast atomic coherent transmission in flying atoms. Finally, we review the research on precision sensing using non-Hermitian exceptional points of \begin{document}${\cal {PT}} $\end{document} -symmetric systems. Near the exceptional points, the eigenfrequency splitting follows an \begin{document}${\varepsilon }^{\tfrac{1}{N}}$\end{document} -dependence, where the \begin{document}$\varepsilon$\end{document} is the perturbation and \begin{document}$ N $\end{document} is the order of the exceptional point. We review the \begin{document}${\cal {PT}}$\end{document} -symmetric system composed of three equidistant micro-ring cavities and enhanced sensitivity at third-order exceptional points. In addition, we also review the debate on whether exceptional-point sensors can improve the signal-to-noise ratio when considering noise, and the current development of exceptional-point sensors, which is still an open and challenging question.
The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm "ghost" state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C operator for the Lee Model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.
非厄米物理是近年来快速发展的重要前沿研究领域,揭示了诸多新奇物理现象与功能应用.然而,当前非厄米物理的研究多集中于经典系统,非厄米特性对量子效应的影响作为亟待探索的关键科学问题,已发展为非厄米物理与量子物理交叉领域的新兴研究方向.无耗散二次型玻色系统中的压缩作用,既能使系统动力学演化呈现等效非厄米特性,又能诱导系统产生量子关联效应.因此,该系统不仅为实现量子体系中多样化的非厄米动力学提供了天然载体,更为深入探究非厄米动力学与量子关联等效应的内在联系、实现基于非厄米特性的量子调控提供了重要平台.本综述将介绍二次型玻色系统中实现非厄米动力学的物理原理,回顾非厄米动力学诱导的典型物理效应,并总结基于非厄米动力学调控系统量子关联的近期研究进展.