1. Introduction

One of the basic assumptions of quantum mechanics is that the Hamiltonian has Hermitian symmetry, which ensures that the energy eigenvalues of the system are real and have observable physical significance. Hermiticity naturally ensures the unitarity of the time evolution of an isolated quantum system, so that the distribution probability of the quantum state in which the system is located is conserved.However, on the one hand, there are few isolated systems in nature, and there are generally open systems with energy flow, particle flow and information flow exchange with the external environment, and the quantum state distribution is no longer conserved. When dealing with such specific problems, the introduction of non-Hermitian Hamiltonian will bring great convenience. ^[1-15], on the other hand, the requirement of Hermiticity is too strong, and the non-Hermitian Hamiltonian can still ensure that the energy eigenvalue of the system is real under certain conditions. ^[16-25].Therefore, it is necessary to develop the non-Hermitian quantum theory. Usually, the energy eigenvalues of non-Hermitian systems are complex numbers, which can be used as an extension of the traditional quantum mechanics theory in the complex field.Non-Hermitian systems are closely related to open and dissipative systems. They have many novel properties, phenomena and applications, and have attracted wide research interest. Typical non-Hermitian Hamiltonians include parity-time symmetric Hamiltonians and pseudo-Hermitian symmetric Hamiltonians.These Hamiltonians have attracted much attention with a wide variety of characteristics, such as singular ^[26-30], Novel Optical Properties ^[31-37] and the study of the dynamics of entropy and the uncertainty relation of entropy for typical non-Hermitian systems ^[38-40], etc.

Quantum simulation was first developed by physicist Feynman. ^[41] was proposed in 1982 and has attracted sustained attention since then. Quantum simulation has now become a powerful tool for scientific research in practice.Quantum simulation can be used to study not only Hermitian quantum mechanical systems ^[42-46], also for studying and simulating non-Hermitian systems ^[38,47-57] and its related novel quantum phenomena provide effective methods.Especially for the latter, quantum simulation has become the main method of quantum-level experimental research. Quantum simulation can construct, operate and observe non-Hermitian systems in the space of controllable quantum systems.So far, this hot frontier has received much attention, and various studies have been carried out on non-Hermitian quantum systems.

In this paper, we briefly introduce the recent progress in the theoretical and experimental studies of quantum simulation of non-Hermitian systems.

2. Development of non-Hermitian quantum systems.

In quantum mechanics, to ensure that the energy levels are real positive and the system is unitary, the Hamiltonian operator of the system H is required to be Hermitian symmetric, i.e.

Where the symbol "†" is the transpose conjugation operator, which represents the combined operation of matrix transpose and complex conjugation.

Pauli in the 1940s ^[1] and Dirac ^[2] proposed non-Hermitian theories to solve the divergence problem of some physical phenomena. Since then, non-Hermitian quantum mechanics has begun to appear.In traditional quantum mechanics (Hermitian Hamiltonian), in order to ensure that physical quantities are observable, the Hamiltonian of a system is required to have Hermitian symmetry, which makes the time evolution of the system unitary, and the sum of the probabilities of the possible quantum States of the system is normalized and does not change with time.However, non-Hermitian quantum systems do not have these characteristics. Not all eigenvalues of non-Hermitian Hamiltonians are imaginary, and there is a special case, that is, non-Hermitian Hamiltonians with parity-time (PT) symmetry, whose eigenvalues may be real. ^[16-19].This discovery has aroused widespread research interest in the scientific community. Bender et al. ^[16-19] and Mostafazadeh ^[20-24] made a systematic study. They constructed many non-Hermitian Hamiltonian models with real eigenvalues and established a complete theory of non-Hermitian quantum mechanics.

2.1 PT -symmetric quantum mechanics.

The field of PT-symmetric quantum mechanics was developed by Bender and Boettcher of Washington University in St. ^[16] was established in 1998.Their research shows that when the Hamiltonian satisfies certain symmetries, its eigenvalues can be ensured to be real, not just Hermitian, so it is possible to describe the physical processes of nature with non-Hermitian Hamiltonians.The aforementioned symmetry is parity-time symmetry, and its Hamiltonian H is expressed by the following equation:

It can be abbreviated as ${\boldsymbol{H}}={\boldsymbol{H}}^{{\boldsymbol{PT}}}$, where P is the parity operator, T is the time evolution operator, and the two commute with each other. The main idea of PT symmetry theory is to replace the Hermiticity of the Hamiltonian with a less constrained PT symmetry.The eigenvalues of a non-Hermitian Hamiltonian with PT symmetry can still be real. Bender The PT symmetric quantum theory of ^[16] is an important part of non-Hermitian quantum theory, which provides a broad space for the development of quantum mechanics.So far, PT-symmetric non-Hermitian quantum systems have attracted much attention in many fields, such as basic theory of quantum mechanics, mathematical physics, open quantum systems, disordered systems, optical systems with complex refractive index and topological insulators. ^{[26,27,58-63]}.

2.2 Pseudo-Hermitian quantum mechanics

After the establishment of PT -symmetric quantum theory, there are still many non-Hermitian systems with real eigenvalues ^[21,64] is not included, and these Hamiltonians do not have PT symmetry. Therefore, PT symmetry is neither a sufficient condition nor a necessary condition for non-Hermitian systems to have real eigenvalues.Such systems with real eigenvalues but not Hermitian symmetry are called pseudo-Hermitian systems. Pseudo-Hermiticity is also a necessary and sufficient condition for the Hamiltonian of the system to have real eigenvalues. Pseudo-Hermitian theory was originally developed by Pauli. ^[1] is built with an indefinite inner product, and its Hamiltonian is

Among, η is usually required to be a linear Hermitian operator.Mostafazadeh, University of Kochi, Istanbul In 2002, ^[20-24] explained the pseudo-Hermitian quantum mechanics system by using biorthogonal basis, and established a complete biorthogonal basis and discrete energy spectrum from the mathematical point of view. η Pseudo-Hermitian quantum mechanical theory.Biorthogonal basis vectors are used to endow the system with completeness and positive definite inner product, but the completeness and positive definite inner product here are different from those in ordinary quantum mechanics. The two sets of eigenfunctions for inner product operation correspond to the Hamiltonian of the system and its Hermitian conjugate operator, respectively.

2.3 PT symmetric, PT antisymmetric, P pseudohermitian symmetric and P pseudohermitian antisymmetric two-level quantum systems

In order to introduce several non-Hermitian quantum systems simulated by linear combinations of unitary operators, this section briefly describes PT-symmetric, PT-antisymmetric (APT), P-pseudohermitian symmetric (PPH) and P-pseudohermitian antisymmetric (APPH) two-level quantum systems.Pick the parity operator P is $\begin{pmatrix}0&1\\1&0\end{pmatrix}$, the time-reversal operator The action of T is to take the complex conjugate, from which the Hamiltonian of a PT symmetric two-level quantum system can be deduced as

Among them, the parameter $r, s, w$ and θ are all real numbers. In a two-level quantum system, the eigenvalue of a PT-symmetric Hamiltonian is $\varepsilon_{\pm}=r\cos\theta\pm \sqrt{w^2+s^2-r^2\sin^2\theta}$, the energy level difference of the system is

$\varDelta_{\text{PT}}$ can only be real or purely imaginary. The PT antisymmetric Hamiltonian follows ${\boldsymbol{H}}=-({\boldsymbol{PT}})^{-1}{\boldsymbol{HPT}}$, which can be obtained by multiplying the PT symmetric Hamiltonian by an imaginary number I, so that the PT antisymmetric quantum system is closely related to the PT symmetric quantum system.In the optical system, the lossless propagation of the PT system corresponds to the refraction-free propagation of the APT system, which makes it possible to manipulate light and form complementary probes in non-Hermitian systems. ^[65].The Hamiltonian of the PT antisymmetric two-level system is as follows:

Eigenvalue of the APT system. Eigenvalues of $\varepsilon_{\pm}$ and PT symmetric systems $\text{i}\varepsilon_{\pm}$ differs by an I factor, so its energy range also has the following relationship:

When the eigenvector of the Hamiltonian of the system is changed under the PT symmetry operator, the Hamiltonian of the PT symmetric system and the PT antisymmetric quantum system is spontaneously broken.However, when the eigenvectors of the Hamiltonian of the system are invariant under the PT symmetry operator, the Hamiltonians of the PT symmetric system and the PT antisymmetric quantum system are unbroken.The spontaneous symmetry broken phase (unbroken phase) of a PT -symmetric system corresponds to the energy range difference $\varDelta_{\text{PT}}$ is the case of pure imaginary (real), while the spontaneous symmetry broken phase (unbroken phase) of the APT system corresponds to the energy range difference $\varDelta_{\text{APT}}$ is the case of real (pure imaginary).When the energy range of the system is zero, the system is at the junction of the broken and unbroken PT symmetry, which is called the phase transition point (EPs). At this time, the energy levels of the system merge and many novel properties appear nearby.In a two-level quantum system η operator with parity operator P The Hamiltonian of a P pseudo-Hermitian symmetric two-level quantum system has the following form:

Among, u and v is real, and the rest of the parameters are the same as above. The eigenvalue of the PPH Hamiltonian in a two-level quantum system is $\varepsilon'_{\pm}=r\cos\theta\pm \sqrt{vu-r^2\sin^2\theta}$, the energy level difference of the system is

$\varDelta_{\text{PPH}}$ can only be real or purely imaginary. In view of the development of PT-symmetric, pseudo-Hermitian, and PT-antisymmetric Hamiltonian quantum systems, and considering the relationship between PT-symmetric and its antisymmetric Hamiltonian, researchers naturally extend to the study of pseudo-Hermitian antisymmetric Hamiltonians.Similar to the relationship between PT symmetric and APT Hamiltonians, the pseudo-Hermitian antisymmetric Hamiltonian is the multiplication of an imaginary number I on the pseudo-Hermitian symmetric Hamiltonian, which satisfies ${\boldsymbol{H}}^†=-{\boldsymbol{\eta}} {\boldsymbol{H}}{\boldsymbol{\eta}}^{-1}$. The η operator is taken as the parity operator, then the Hamiltonian of the corresponding APPH two-level quantum system is as follows:

The eigenvalues of the APPH system are $\text{i}\varepsilon'_{\pm}$, the energy level difference is

Phase transition points also appear in the PPH and APPH quantum systems, namely $\varDelta_{\text{PPH}}$ and $\varDelta_{\text{APPH}}$ is equal to 0. Taking the phase transition point as the boundary, the PPH system and the APPH system can be divided into two phases: the real energy range and the virtual energy range.Fig. 1 ^[38] and Fig. 2 ^[38] gives the three parameter spaces of PT symmetric, APT, PPH and APPH systems, and describes the phase transition points EPs, the phase space and the internal relations of the parameter spaces.

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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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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3. Theory of quantum simulation for non-Hermitian systems based on LCU

The concept of quantum simulation was first proposed in the 1980s to simulate a quantum system with another controllable system. ^[41,66]. Since then, quantum computing has become one of the most fruitful areas of research in physics.Quantum computer uses quantum properties such as quantum superposition state and quantum entanglement to process information, and its powerful parallel function makes the computing speed exponentially fast. However, the function of quantum computing needs to be embodied by quantum algorithm.Benioff ^[66] proposed the concept of quantum computation and studied it in 1980, Feynman ^[41] proposed the simulation of quantum systems by quantum computers in 1982, Deutsch ^[25] perfected the concept of quantum computation in 1985.But until 1994, Shor ^[67] Proposed Decomposition of Large Numbers, 1996 Grover After the quantum search algorithm was proposed in ^[68], quantum computing has attracted wide attention from the scientific community and has become an international research frontier that continues to this day. ^[69].Traditional quantum algorithms process information through the unitary evolution of the system, and a calculation process is equivalent to a dynamic evolution of a closed system.In recent years, various methods of non-unitary time evolution have emerged. ^[14,70-78], including linear combinations of unitaries (LCU) ^[14,72,73], imaginary time evolution method ^[74,75], Time-dependent variational method ^[76] and extension method ^[77,78].

3.1 LCU-Based Quantum Simulation Method

In 2006, Professor Long Guilu of Tsinghua University ^[72,73] First proposed LCU algorithm.Different from the traditional algorithm which can only carry out multiplication and division of unitary operators, the LCU algorithm can realize the four arithmetic operations of addition, subtraction, multiplication and division of unitary operators, thus realizing non-unitary quantum gate operations with a certain probability, and further expanding the ideas and methods of constructing quantum algorithms.Subsequently, the LCU algorithm has been rapidly developed ^[79-88] has become one of the most powerful tools for designing new quantum algorithms and has been widely used. ^[89].

3.2 Construction of Typical Non-Hermitian Quantum Systems Based on LCU

3.2.1   Quantum simulation of generalized PT -symmetric two-level system

Due to non-Hermiticity, the evolution of a PT -symmetric quantum system is not unitary.Although simulations of some particular PT -symmetric quantum systems have been proposed ^[47,49,90], and realized in the experiment ^[47,49], but it is still a problem for conventional quantum computers to simulate generalized PT symmetric non-Hermitian quantum systems.Literature [ 51] theoretically studied generalized PT-symmetric two-level quantum systems, which are applicable to both PT-symmetric systems (the off-diagonal elements are complex conjugates) and PPH-symmetric systems (the off-diagonal elements are all real numbers).Using the idea of LCU, the researchers first constructed a PT symmetric Hamiltonian system in a four-dimensional Hilbert space, and then simulated its time evolution. ^[51].This experiment uses the generalized PT symmetric Hamiltonian and the auxiliary qubit to promote the evolution of the working qubit, and realizes the fast evolution of the generalized PT quantum short-time problem (such as Fig. 3 ^[51]).This simulation method provides theoretical support for the quantum simulation of generalized PT symmetric two-level systems in nuclear magnetic resonance quantum systems and quantum optical systems.In addition, researchers have shown how to realize generalized PT -symmetric two-level systems with qutrit and qudit as auxiliary qubits, respectively (such as Fig. 4(a) and Shown in Fig. 4(b) ^[51]), in some cases, these systems have advantages over quantum computers with qubits as auxiliary qubits.

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

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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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3.2.2   Quantum simulation of generalized PT antisymmetric two-level system

In addition to the PT symmetric system, the PT antisymmetric system ^[91,92] has also attracted much attention because of its novel properties and potential applications, such as optical materials with APT photonic structures with balanced positive and negative refractive indices. ^[93], APT optical system with constant refraction ^[94], diffusion system with APT property ^[95], coupling-induced unitary and non-unitary scattering in APT systems ^[96], etc.Therefore, the simulation of APT quantum system is particularly important.For the first time, researchers have proposed a theoretical scheme based on LCU quantum simulation of APT two-level system. ^[55], the proposed quantum simulation theory scheme is applicable to both APT systems (the off-diagonal elements are complex conjugates of each other) and APPH systems (the off-diagonal elements are all real numbers).It is found that the minimum Hilbert space dimension required by the scheme is six, and the general task can be accomplished by a quantum computer composed of a qutrit and a qubit.Using the dual quantum algorithm ^{[72,73,80,81]}, a quantum circuit of a qubit-qutrit hybrid system is designed, which can realize the LCU. Qutrit as an auxiliary bit for constructing a generalized apt subsystem.When qutrit is in state When measured in $|0\rangle_{a}$, the generalized PT anti-symmetric Hamiltonian operator will push the evolution of the working bit (such as Shown in Fig. 5 ^[55]).In addition, researchers have designed qubit systems to implement the quantum simulation process, giving the flow chart and quantum circuit of qubit quantum computer (such as Shown in Fig. 6 ^[55]), both the flow chart and the quantum circuit are designed for a three-qubit quantum computer, making it possible to use existing technology for experimental implementation.

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

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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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3.2.3   Quantum simulation of PT arbitrary phase symmetric systems.

For the first time, researchers have generalized PT symmetric and PT antisymmetric systems to PT arbitrary phase (PT- φ) symmetric system ^[97,98].For a non-Hermitian Hamiltonian H, researchers found the PT symmetric and PT antisymmetric Hamiltonian ${\boldsymbol{H}}_{\text{PT}}$ and ${\boldsymbol{H}}_{\text{APT}}=\text{i}{\boldsymbol{H}}_{\text{PT}}$, such that H can be viewed as ${\boldsymbol{H}}_{\text{PT}}$ and The combination of ${\boldsymbol{H}}_{\text{APT}}$ is shown in the following formula:

Among φ is the phase associated with the symmetry of the Hamiltonian, ( 12) is equivalent to ${\boldsymbol{H}}=\cos\dfrac{\varphi}{2}{\boldsymbol{H}}_{\text{PT}}-\sin\dfrac{\varphi}{2}{\boldsymbol{H}}_{\text{APT}}$. This expression has the same commutation form as an anyon, so it is called H is PT arbitrary phase symmetric or PT- $\varphi$ symmetric Hamiltonian (also called anyonic PT symmetric Hamiltonian in some articles).And the Hamiltonian H Symmetry-dependent phase φ is fixed to $2 k\pi$( k takes integer), the system is the PT -symmetric case; the phase is fixed as $(2 k+1)\pi$, the system is PT antisymmetric ( k (integer).This method of representing non-Hermitian Hamiltonians is applicable to quantum systems in arbitrary dimensions. 99] gives the general form of the Hamiltonian in two dimensions and discusses its basic properties, such as eigenvalues, the condition of whether the PT symmetry is spontaneously broken, etc.At the same time, researchers have theoretically studied the quantum simulation of the time evolution of PT arbitrary phase symmetric two-level system with traditional Hermitian system, and designed the qubit- qudit hybrid based on LCU (such as Shown in Fig. 7 ^[99]) and pure qubits (such as Fig. 8 ^[99]) quantum circuit of the device.Both schemes consist of a working qubit and an auxiliary qubit (or qudit). After a series of quantum gate operations, the working qubit evolves in time according to a non-Hermitian Hamiltonian with a certain probability.The former scheme clearly demonstrates the simulation method, while the latter scheme is more practical and has a higher probability of success. These two schemes are expected to be experimentally realized in small quantum devices, such as nuclear magnetic resonance and quantum optical systems.Similar to PT arbitrary phase-symmetric non-Hermitian systems, the quantum simulation theory of pseudo-Hermitian arbitrary phase-symmetric systems is described in [ 100].

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

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3.2.4   Quantum simulation of P pseudohermitian antisymmetric two-level system

Literature [ 101] studied the quantum simulation theory of a P-pseudo-Hermitian antisymmetric two-level system. The researchers used the dual quantum computing method to effectively simulate the evolution of an APPH system from an arbitrary initial state to different phases.Three qubits are essential for quantum simulation of arbitrary APPH systems in a qubit computer. Researchers have found that six-dimensional and eight-dimensional (such as Fig. 9 and Shown in Fig. 10 ^[101]) schemes can all be simulated, but each has different advantages.The six-dimensional scheme has a higher probability of success, while the eight-dimensional scheme requires fewer quantum gate operations. Therefore, the choice of the scheme depends on the stability and controllability of the experimental system. When the APPH system is in some special phases, it can be simulated with fewer qubits.For example, when the off-diagonal elements of the system Hamiltonian parameters are equal, only two qubits are needed to simulate, and the success probability is higher.

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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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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3.2.5   Quantum simulation of τ-pseudohermitian antisymmetric two-level system

In addition to the PT-symmetric system, the P-pseudohermitian symmetric system and their corresponding antisymmetric systems discussed above, Mostafazadeh, 2002 ^[22] defined and studied a class of Hamiltonians as τ-pseudohermitian antisymmetric ( Non-Hermitian system of τ-APH).The τ-APH Hamiltonian satisfies

Among, τ is an antilinear antihermitian invertible transformation. The τ-APH Hamiltonian and its operators τ is of great significance for the further study of the necessary and sufficient conditions for non-Hermitian Hamiltonians with real energy spectrum.Literature [ 57] studied LCU based ^[72] pair τ-Quantum simulation of pseudo-Hermitian antisymmetric two-level system.It is proposed in detail that the anti-linear operator τ is designated as T operator or P operator later, how to simulate T-APH using three qubits in general and two qubits in special cases (as Fig. 11 ^[57]) and PT-APH (such as Shown in Fig. 12 ^[57]).In general, the smallest Hilbert space needed to simulate time evolution using LCU is six dimensions.In special cases, both schemes can be simulated, but the two-qubit scheme has a higher probability of success, depending on the initial state, the Hamiltonian, and the dimension of the Hilbert space used.Therefore, it is of great significance to combine the unitary extension term based on LCU and the phase matching condition for the required number of dimensions before quantum simulation, saving the source of quantum bits and improving the success probability.This simulation method can be extended to simulate general τ-APH high level system, which can be realized experimentally in the future.

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

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

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4. Quantum simulation experiment based on LCU for non-Hermitian system

The theoretical scheme of quantum simulation based on LCU for non-Hermitian systems has been introduced before, and now the experimental scheme is introduced. Quantum simulation experiments can use different systems, such as nuclear magnetic resonance quantum systems, quantum optical systems, superconducting quantum systems, ion trap systems, ultracold atomic systems, etc.In this section, the experimental scheme of quantum simulation of non-Hermitian system using LCU method in NMR quantum system and quantum optical system is introduced.

4.1 Experimental demonstration of digital quantum simulation for generalized PT -symmetric systems

Digital quantum simulation (DQS) is an important experimental method for quantum simulation, which can be used as a universal quantum computer ^[85]. The DQS algorithm is used to simulate a quantum system with a circuit model whose quantum operations are decomposed into a time-ordered sequence of general quantum gates.The researchers based on the literature [ The quantum simulation method for generalized PT symmetric systems proposed by 51] demonstrates a DQS general method for implementing generalized PT symmetric operators using circuit models in the framework of quantum computation. ^[52].The entanglement properties of PT symmetric systems were experimentally studied on the NMR platform, and the application of the scheme in entanglement retrieval was demonstrated.It should be emphasized that entanglement recovery in general non-Hermitian operators can be achieved by appropriate modification of the original quantum circuit, which means that general two-level system evolution can be achieved without Hermitian restriction by the protocol demonstrated in the experiment.This experiment reveals the oscillation of the entropy and entanglement of two qubits in the unbroken phase of PT symmetry in a bipartite non-Hermitian system.

4.2 Experimental demonstration of a non-Hermitian system with non-zero entropy stable state under PT symmetry breaking

The study of Section 4.1 is about a bipartite non-Hermitian system in which two qubits (Alice and Bob) are initially entangled and Alice evolves under a local PT symmetric Hamiltonian.Such a two-qubit model would lead to oscillations of entropy and entanglement in the PT-symmetry unbroken phase, which violates entanglement monotonicity ^[52,86]. Specifically, the entropy and entanglement of the two qubits in the broken phase decay exponentially to zero and form a stable state that does not change with time.The dynamic process of this stable state is called normal dynamic mode (NDP), which is only related to the quantum phase, but not to the degree of non-Hermiticity.

However, the literature [ 87] found that when the system is extended from the bipartite model to the tripartite model, another evolution process, called the anomalous dynamic pattern (ADP), occurs. The evolution of entropy and entanglement in a three-qubit system with local PT symmetry is studied theoretically and experimentally.Two dynamical modes, ADP and NDP, are found in this system, where the entropy and entanglement tend to stabilize at non-zero values associated with non-Hermitian in ADP, which is absent in bipartite systems.The bipartite subsystem in ADP exhibits the maximum increase of entanglement at the singular point, and the mutual information can exceed the initial value. In addition, the experimental demonstration of stable States in non-Hermitian systems with non-zero entropy and entangled States is realized on a four-qubit quantum simulator with nuclear spins.When the PT symmetric system is extended from bipartite to tripartite, some different physical properties will appear, and the enhancement of entanglement and mutual information has important physical significance.

4.3 Experimental simulation of PT -symmetric dynamics using photonic qubit

The simulation and application of PT symmetry based on classical optical experimental techniques are mature. However, there are still challenges in experimentally constructing and studying PT symmetric systems using linear quantum optics.Researchers based on the theory of LCU non-Hermitian quantum simulation ^[14,72,73], the quantum simulation of a generalized PT symmetric system is realized by using a linear quantum optical system. ^[102].Such as Fig. 13 As shown in ^[102], the system is amplified by using an auxiliary qubit, and the all-Hermitian system subsystem is simulated by a post-selection process. The phase of the $U_{\text{PT}}$ operator. It is shown that the state in the evolution process can be observed with high fidelity when only the PT symmetric evolution subspace is considered.Due to the efficient operation of the extended method, this work provides a way to further exploit the singularity of PT symmetric Hamiltonians for quantum simulation and quantum information processing.

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

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4.4 Experimental observation of information flow in a PT antisymmetric system of nuclear spins

Based on the literature [ According to the theory proposed by [55], the researchers first experimentally simulated the time evolution of a generalized PT antisymmetric system on a nuclear magnetic resonance quantum computing platform, and found that the information backflow in the PT symmetric system also exists in the PT antisymmetric system, and the experiment demonstrated the oscillation of the information flow. ^[56].Experiments show that in the broken phase, the information flow oscillates back and forth between the environment and the system, and the phenomenon of information backflow occurs (such as Shown in Fig. 14 ^[56]), which does not exist in conventional Hermitian quantum mechanics.The results also show that the period and amplitude of the oscillation increase monotonically when the system parameter approaches the singularity before the phase transition, and the information in the environment will not flow back to the system after passing through the critical point, which realizes the symmetry breaking phase transition process of the PT antisymmetric system.The monotonic correspondence in the symmetry-breaking phase shows that this result can provide a measure of non-Hermiticity for quantum systems.In addition, the researchers also found an interesting inverse correspondence: when the PT symmetric system is in the broken phase, there is no information flow oscillation or information backflow, while in the broken phase of the PT antisymmetric system, there is information backflow.

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

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5. Other quantum simulation theory schemes.

In this section, we briefly introduce other theoretical schemes for quantum simulation of non-Hermitian systems, including quantum random walk, embedded quantum simulation, and space extension.

5.1 Quantum random walk

1993, Aharonov et al. ^[103] first proposed the concept of quantum random walk, which is the contrast of classical random walk in the quantum world.There are two forms of quantum random walk: discrete time quantum walk (DTQW) and continuous time quantum walk (CTQW).The main difference between the two forms is the time consumed when using the evolution operator. In DTQW, the corresponding evolution operator for the system acts only on discrete time steps, while in CTQW, the evolution operator can be used at any time.Where the continuous quantum random walk is given by Farhi and Gutmann ^[104] was first proposed in 1998, while discrete quantum random walks were introduced by Watrous ^[105] was proposed in 2001.

Quantum random walk has a wide range of applications in quantum information, such as the search algorithm for disordered databases developed by quantum random walk. It is precisely because of the characteristics of quantum random walk superior to classical random walk that the diffusion rate of quantum States carrying information increases quadratically compared with classical random walk.In the past, researchers generally believed that ballistic diffusion was a characteristic of quantum random walks and took advantage of it.But in 2015, Professor Xue Peng's research group For the first time, the propagation, diffusion and recovery of light information in quantum random walk were observed in ^[106]. This theoretical and experimental verification of the dynamic evolution of light in quantum random walk has subverted people's understanding of quantum random walk, provided a new direction for the application of quantum random move in quantum information, provided a new perspective for understanding the basic phenomena of dynamic evolution based on quantum mechanics, and also provided a new perspective for the study of quantum diffusion and quantum topological phenomena.Then in 2017, Professor Xue Peng's research group ^[91] experimentally realized PT symmetric quantum random walk in an open system for the first time and observed a new one-dimensional topologically protected boundary state, which provides a new basis for quantum computation based on quantum random walk platform.The research group designed an open system quantum walk model, which replaces the loss-gain that is difficult to achieve with the Walker's probability alternating loss-non-loss, and proves that it also satisfies PT symmetry.The quantum walk of PT symmetry is realized by a single photon in a linear optical system, and the quantum properties of PT symmetry preservation, breaking and critical point are demonstrated respectively. The dynamic evolution process of a real PT symmetric quantum system is realized for the first time.

5.2 Embedded quantum simulation

Solano Group, University of the Basque Country, Spain ^[107-109] introduced the concept of embedded quantum simulation for the first time.Embedded quantum simulation refers to the introduction of auxiliary qubits to compile the mechanical quantities and Schr Schrödinger equations in the original space into an amplified Hilbert space, which corresponds to the current highly controllable quantum simulation platform. After time evolution, it is back-projected back to the original space to realize the observation of non-physical operations that violate causality.Embedded quantum simulators enrich the content of quantum simulation and realize phenomena that can not be observed in nature.Currently, embedded quantum simulators have been applied to simulate complex conjugation of wave functions, antilinear and antiunitary operations ^[107] and its experimental observation on trapped ion platform and optical platform ^[110,111], to study entanglement monotonicity in quantum computation ^[109] and its verification scheme using MΦlmer-SΦrensen quantum gate and local rotation gate in trapped ion platform experiment ^[112].In recent years, the simulation of PT symmetric systems has been achieved using embedded methods. ^[113-115].This embedded simulation method is based on Naimark's amplification theory, which amplifies a non-Hermitian PT symmetric Hamiltonian to realize a higher dimensional Hermitian Hamiltonian, and implements a fixed projection operator on the post-selected auxiliary bits.However, the success probability of this kind of embedded simulation of PT-symmetric systems can be improved. A local operator and classical communication (LOCC) protocol is proposed to simulate the dynamical evolution of PT-symmetric systems. ^[88], only one auxiliary bit is needed.In general, the success probability is more than twice that of the original embedding method, and in special cases, the success probability can even approach 100%. Moreover, this LOCC protocol is more flexible, less dependent on the metric operator, and more suitable for particle applications.Compared with LCU simulation of non-Hermitian systems requiring at least one auxiliary qubit, LOCC embedded quantum simulation requires only one auxiliary qubit to realize PT symmetric systems.

5.3 Spatial expansion

Günther and Samsonov, Ruhr University, Germany In 2008, ^[90] proposed the Naimark dilation of a short-time quantum system with a PT-symmetric Hamiltonian, which is reinterpreted as a subsystem of a Hermitian system in a high-dimensional Hilbert space.This is a direct experimental realization of Bender et al. In an entangled two-spin system. The PT-symmetric ultrafast short-time problem proposed by ^[116] opens the way. It solves the switching problem between the quantum mechanics of the PT-symmetric Hamiltonian and the traditional Hermitian quantum mechanics mode.The key idea is to reinterpret the brachistochrone as an appropriate symmetric subsystem of a larger Hermitian quantum mechanical system in a higher dimensional Hilbert space.Based on Naimark expansion technique The large system generated by ^[117] will have the structure of an entangled two-spin (two-qubit) system, so the experimental realization of short-time effects is feasible.

6. Other quantum simulation experimental systems

So far, nuclear magnetic resonance quantum systems, quantum optical systems, superconducting quantum systems, ion traps, ultracold atoms and other systems have become popular alternatives to construct universal quantum processors because of their different advantages.In this section, we will introduce quantum simulation experiments in nuclear magnetic resonance systems, quantum optics and photonics.

6.1 Nuclear magnetic resonance system

NMR has the advantages of long coherence time, precise pulse manipulation and high fidelity, which plays a vital role in the study of quantum information. ^[118].As a quantum simulator, the NMR system can be used to simulate the basic quantum mechanical model, quantum delayed choice ^[119,120], Quantum Phase Transition ^[121,122], quantum tunnelling ^[47,123] and other quantum systems that are not easily manipulated.Several specific applications of quantum simulation using NMR systems are described below.

6.1.1   Observation of fast evolution of PT -symmetric systems.

The brachistochrone problem describes the shortest time evolution between two States. In quantum mechanics, the brachistochrone between two States is bounded by the maximum difference between the eigenvalues of the Hamiltonian, which can be applied to time-optimal methods of quantum algorithmic complexity.Bender et al ^[116] It has been demonstrated that PT-symmetric Hamiltonians have a faster minimum time of evolution than Hermitian quantum mechanics.In nuclear magnetic resonance (NMR) quantum systems, experiments simulating the time evolution of PT-symmetric Hamiltonians have been designed and implemented. ^[47].The experimental results show that the evolution speed of the PT symmetric Hamiltonian system is indeed faster than that of the Hermitian quantum system, and the evolution time can be close to zero. In Hermitian quantum mechanics, when the difference between the eigenvalues of the Hermitian two-level quantum system is fixed, the fastest evolution time of the spin flip is unchanged.For PT symmetric quantum systems, the brachistochrone time can be changed by changing the parameters in the Hamiltonian, as Bender et al. As predicted by ^[116] and demonstrated in the work, changing the parameters of the Hamiltonian can not only speed up the evolution, but also slow it down. ^[124].

6.1.2   Quantum simulation experiment of avian compass in NMR system

Quantum biology is a new field that combines quantum science with biology. It studies quantum effects in living systems and explores the role of quantum effects in biology.The phenomenon of avian magnetoreception first gained experimental support in the 1960s ^[125], it is the ability of birds to orient in the Earth's magnetic field through a quantum compass.After experiments on European robins, it was suggested that the compass might rely on a pair of electron spins in a radical pair to interact with the Earth's magnetic field through Zeeman interaction.Other experiments have shown that the chemical compass is not affected by polarity reversal ^[126], but will be destroyed by an RF field with a frequency near the resonance frequency of the pair ^[127,128], and only when there is ambient sunlight and the magnetic field strength is between $\pm30$% range of internal magnetic field to work properly ^[129,130].

One class of quantum simulation system is based on NMR of spins in certain compounds. Each chemical substance usually has a fixed number of nuclear spins, which are manipulated by researchers using the natural Hamiltonian within the NMR spectrometer and by applying carefully tailored RF pulse sequences.Where the natural Hamiltonian originates from the static magnetic field and spin coupling in the chemical ^[131]. This work studies an open system in which the system is coupled to an external environment, so that the Hamiltonian of the system is non-Hermitian. The RF pulse matches or nearly matches the Larmor frequency of the nucleus in the compound.In this case, the nucleus can easily absorb and emit the energy imparted by the incident RF pulse, thus contributing to the precise control of the nuclear spin. Therefore, the simulation of the bird compass can be realized by using such a quantum simulation system.

Literature [ 44] describes an experimental demonstration of avian magnetic induction in a nuclear magnetic resonance quantum information processor. First, prepare a $|00\rangle$ state and then apply a series of quantum logic gates to create a singlet state. Apply the time evolution operator of the bird compass to the singlet state.Finally, the same sequence of quantum logic gates returns the system to $|00\rangle$ state, the researcher reads the data in this quantum state. Both the logic gate sequence and the time evolution operator can be implemented using the gradient ascent optimization algorithm (GRAPE).Experiments show that the reference and detection compass models are successfully simulated in the NMR system and are in good agreement with the theory, so the NMR quantum simulation is widely applicable.

6.1.3   Quantum spacetime on a quantum simulator.

Quantum gravity aims to unify Einstein gravity with quantum mechanics, so that people's understanding of gravity can be extended to the Planck scale. $1.22\times10^{19}\;{\rm{GeV}}$. At the Planck scale level, Einstein gravity and the spacetime continuum are decomposed and replaced by quantum spacetime.Many current approaches to quantum spacetime are rooted in spin networks, an important, non-perturbative framework for quantum gravity. Spin networks are Penrose ^[86] was inspired by the theory of torsors and later widely used in Loop Quantum Gravity (LQG). ^[132].In LQG, spin networks represent quantum States of the underlying discrete quantum geometry of space at the Planck scale, used as certain Boundary data of $3+1$-dimensional quantum spacetime. Quantum spacetime can have an open system structure and is therefore closely related to non-Hermitian systems.

Such as Shown in Fig. 15 ^[133], a spin-network bounded $3+1$-dimensional quantum spacetime is a spin foam, a "network" of many three-dimensional world graphs (surfaces) and their intersections, where the world graphs are colored by spin halves.Like the classical spacetime formed by the time evolution of a classical space, the time evolution of a spin network forms a quantum spacetime ^[134,135].

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]

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Researchers have used quantum simulators to explore spin network States and spin foam amplitudes, providing an effective experimental demonstration for studying LQG ^[133].By using a four-qubit quantum register in a nuclear magnetic resonance system, created a Fig. 16 10 invariant tensor States representing quantum tetrahedra in ^[133] with fidelity exceeding 95%.Then, using these quantum tetrahedra, a spin in the Ooguri model is simulated Spin Foam Vertex Amplitude for $-1/2$.Since the vertex amplitude determines the spin foam amplitude and displays the local dynamics, the result shows an interaction amplitude of 5 glued quantum tetrahedra or m to $(5-m)$ Transitions of quantum tetrahedra.

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

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6.2 Quantum optical system

Since the emergence of quantum mechanics, optics has played a very important role in quantum physics and is closely related to the development of quantum. Therefore, quantum optical systems have become one of the key alternatives for quantum computation and quantum information processing.

6.2.1   Direct experimental simulation of the Yang-Baxter equation

Because of the importance of the Yang-Baxter equation (YBE), its experimental verification has always been the goal of people.It is noteworthy that Tennant et al. ^[136,137], they measured the spectrum of Heisenberg half-spin chain, and the experimental results are consistent with the calculation results of YBE.Literature [ 138] The measured density distribution of one-dimensional wire is in good agreement with the theoretical calculation based on Young's solvable model. However, these experiments are only indirect verification of YBE.The fundamental reason is that YBE only provides sufficient conditions for the spectrum or the observed profile, that is, the observed profile is only a necessary condition for YBE, which does not guarantee the validity of YBE, nor can it prove the Lorentz-like transformation of the spectral parameters.

The first direct experimental simulation of YBE using linear quantum optics was reported by researchers in 2013 ^[139].The basic principle of this simulation is given by Hu et al. ^[140] was established in 2008. By using the Temperley-Lieb algebra, they performed a remarkable reduction to obtain a YBE of dimension 2, which makes it possible to implement the YBE in quantum optics with current technology.In the experimental simulation in 2013, the researchers realized the Hu-Xue-Ge scheme with linear quantum optical elements such as beam splitters, half-wave plates and quarter-wave plates. ^[140], and prove the validity of the YBE, and also directly verify the equality of the two sides of the YBE for Hermitian symmetry.Moreover, for the first time, the researchers have experimentally demonstrated the Lorentz-like transformation of the spectral parameters of YBE. However, the simulation mentioned by the researchers does not involve the quantum entanglement problem in YBE, and the realized YBE is Hermitian symmetric.The simulation of non-Hermitian YBE is introduced below, which provides a new idea for realizing the quantum entanglement of YBE. 49] realized the quantum simulation of non-Hermitian YBE based on LCU for the first time. The simulation system contains a Yang-Baxter subsystem and an auxiliary qubit.The Yang-Baxter subsystem will evolve with the parts on both sides of the YBE, and the auxiliary system is used to maintain the unity of the YBE simulation and is a necessary condition for checking its correctness.In the traditional quantum simulation, the two sides of the equality sign of the YBE equation are simulated separately, and then the two output States of the whole Yang-Baxter system are reconstructed and compared in essence to check the correctness of the YBE simulation. In this process, the integrity of YBE is destroyed.On the contrary, in the simulation based on LCU, the two sides of the YBE can be simulated as a whole at the same time due to the effect of the auxiliary bit, so the integrity of the YBE will not be destroyed. Therefore, the quantum entanglement on both sides of the YBe or the quantum entanglement of multiple YBE systems is expected to be realized.Moreover, the auxiliary qubit can be used to detect defects in the YBE simulation process. This process does not require the quantum state of the YBE subsystem to collapse, so the final YBE state can be preserved.In addition to proposing theories, researchers have also proposed experimental schemes to realize YBE on quantum optical systems and nuclear magnetic resonance quantum systems (such as Fig. 17 ^[49]). This scheme can simulate both non-Hermitian and Hermitian YBE systems efficiently.

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

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6.2.2   Optical simulation of quantum heat engine

Heat engines play an important role in the development of human society and life. How to improve the efficiency of heat engines has always been the core scientific issue of thermodynamics. Quantum heat engine (QTE), as its name implies, is a heat engine that uses quantum systems as working substances to do work.Because QTE does work to the outside world and the parameters of the interaction between the system and the outside world are controllable, it is an open system and can be described by a non-Hermitian Hamiltonian. In order to improve the efficiency of QTE, researchers have explored the quantum properties of the working substance and achieved many effective results.Recently, by nuclear magnetic resonance (NMR) Nitrogen vacancy center in ^[141] and diamond ^[142] manipulated nuclear spin system, the researchers experimentally verified the performance of the QTE.In general, a major difficulty in implementing QTE in physical systems is the high degree of controllability required to achieve robustness against decoherence. Therefore, it is particularly important to design QTE in systems where reservoirs can be effectively controlled.In addition, decoherence can be realized by non-unitary operators, and the QTE is associated with a non-Hermitian Hamiltonian.

In order to simulate controllable reservoirs, the effect of quantum channels on quantum information must be taken into account ^[143].In this context, it is crucial for researchers to consider optical implementations of relevant quantum channels, such as amplitude damping, phase damping, and bit-flip channels, as well as other channels performed using single photons ^[144].Literature [ 145] introduced a theoretical and experimental scheme to simulate quantum heat engines by optical methods. The behavior of the working substance and the heat reservoir is realized by the internal degrees of freedom of a single photon.By using polarization and propagation paths, the researchers encoded two qubits and then realized the thermodynamic steps of the Otto cycle. To illustrate the feasibility of the scheme, the researchers realized this simulation through intense laser beam experiments and evaluated the heat and work of each step of the thermodynamic cycle.

6.3 Photonic system

Photonic system has become one of the most ideal and popular physical systems for quantum information processing because of its fastest propagation speed, strong stability and good ability to resist environmental interference.

6.3.1   Observation of critical phenomena in PT -symmetric quantum dynamics

PT-symmetric non-Hermitian systems are novel in synthetic systems (classical optics, microwave cavities, quantum gases, and single-photon systems). In these systems, the spectrum is completely real in the PT-symmetric unbroken phase, in contrast to the case of the PT-symmetric spontaneously broken phase.As a result, the dynamics in the two phases are distinct and dynamical criticality emerges at the boundary between the two regions.A paradigm for PT symmetric non-unitary dynamics in open quantum systems is the reversible-irreversible criticality of the information flow between the system and its environment ^[132].Here, when the system is in the PT-symmetry-unbroken regime, the information lost to the environment can be fully recovered due to the presence of a finite-dimensional entangled part in the PT-symmetry-protected environment. On the contrary, when the PT-symmetric system is spontaneously broken, the information flow is irreversible.In the vicinity of the singularity, the physical quantities show power-law behavior, and a large number of experiments have observed such novel dynamic characteristics and the characteristics of the PT transition point or singularity in the classical PT symmetric system with balanced gain and loss. ^[31-36].

Literature [ 146] described the experimental simulation of PT-symmetric non-unitary quantum dynamics using a single-photon interference network, and experimentally studied the critical phenomena in the information flow near the singular point in the PT-unbroken and broken regimes.By implementing non-unitary gate operations on photons and performing quantum state tomography, the researchers reconstructed the density matrix of the PT dynamics over time at any time, which enabled the characterization of critical phenomena near the PT singularity and demonstrated power-law behavior in key quantities such as distinguishability and recurrence time.Both symmetry and initial conditions have an impact on critical phenomena. By introducing an auxiliary degree of freedom as the environment and detecting the quantum entanglement between the system and the environment, researchers have confirmed that the observed information recovery is caused by the finite-dimensional entanglement in the environment.This work is the first experiment to describe critical phenomena in PT symmetric non-unitary quantum dynamics, and opens up a way to simulate PT symmetric dynamics in synthetic quantum systems.

6.3.2   Analog PT arbitrary phase symmetric system in optical microcavity

Section 3.2.3 has introduced the PT arbitrary phase symmetric system and given its simulation theory based on LCU. This section introduces the simulation of PT arbitrary phase symmetric system in binary optical microcavity.The Hamiltonian of a PT arbitrary phase symmetric system satisfies ( 12), when the phase associated with the Hamiltonian of the system When φ is at different values, a system with PT or APT symmetry can be obtained.Literature [ 97] studied the evolution of PT arbitrary phase symmetric systems, which can be realized in binary coupled systems.Experimentally, two optical microresonators with the same resonant frequency were selected for simulation, and it was found that the energy level degeneracy of the PT arbitrary phase symmetric system can only be achieved in the PT symmetric and PT antisymmetric cases.Researchers have proposed experimentally realizable systems that demonstrate symmetry protection at arbitrary phases of PT. Such systems are expected to be used for hypersensitive sensing, optical chirality, and nonreciprocal transmission.

6.3.3   Simulation of quantum transitions in a three-level system using photonic Gaussian modes

Quantum optical systems are shown to be an effective choice for fundamental tests of quantum mechanics and for the implementation of quantum information protocols.Certain optical systems encode information in the photon transverse momentum, which can be obtained using a slit ^[147] or different photon paths ^[148] to discretize in order to prepare single-, two-, or four-photon quantum States in slit mode or Gaussian mode ^[149,150].Gaussian modes can be used for generalized quantum operations because Gaussian shapes are preserved in sequential operations. In addition, such States are easily coupled to lensed optical fibers and photonic circuits, allowing researchers to explore them on different platforms and in different applications.Another interesting possibility is to use a spatial light modulator (SLM) as part of the optical system to convert the photon state, keeping the same Gaussian encoding, to achieve a wide range of quantum operations ^[150,151]. An important advantage of such optical structures is their ability to simulate complex quantum systems.

Researchers proposed and implemented a method to experimentally simulate a non-Hermitian three-level system under quantum transitions using a three-mode photonic system (as Fig. 18 ^[152]), three different spontaneous decay dynamics are simulated in the three-level atomic system: cascade decay, Λ decay and V decay.Using a photon-level attenuated optical coherence source, the researchers prepared photons in a three-way superposition state, which were encoded in a parallel Gaussian mode. By precisely modulating the periodic phase of the photon path, they were able to perform a large number of operations and simulate different attenuation dynamics in a three-level system.This simulation gives us a better understanding of how quantum transitions affect the coherence of a three-level system. In addition, this implementation can be used to understand how quantum transitions in high-dimensional systems affect quantum protocols due to state decoherence, which can be implemented by non-Hermitian unitary operators and thus be related to non-Hermitian systems.The method is universal and can be extended to multilevel systems with more than three levels.

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

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6.3.4   Photonic quantum simulation of coupled PT -symmetric Hamiltonians

Literature [ 153] proposed a framework for quantum simulation of PT-symmetric Hamiltonians suitable for direct implementation of unitary transformation technology platforms. Researchers used the device to embed non-unitary operators, as well as a second operator with an opposite time evolution operator to the first operator, into a global unitary transformation.The global evolution in this model allows single- or many-particle excitations to tunnel between the coupled systems with a probability proportional to the non-Hermiticity of the analog Hamiltonian. This construction allows for the experimental study of States superimposed in opposite time directions in the case of a non-Hermitian Hamiltonian.Using a programmable integrated photonic chip and a collection of single-, two-, and three-photon input States, the researchers experimentally simulated many-particle dynamics in two- and three-mode PT-symmetric Hamiltonians.The experiment reproduces the dynamics in the unbroken PT symmetry regime and across the singularity to the broken PT symmetry regime, including the mutual coherence and interference effects between the time-forward and time-reverse subsystems.This work shows the possibility of using programmable quantum simulators to study fundamental problems in quantum mechanics.

7. Simulation experiment of non-Hermitian system in classical system

In addition to quantum systems, classical systems can also simulate non-Hermitian systems. Classical physical experimental systems are more mature and stable. Compared with quantum systems, classical physical experimental systems are easier to manipulate and implement, and can also demonstrate the characteristics of non-Hermitian systems.The following is a brief introduction to the simulation of PT symmetric non-Hermitian systems using LRC circuits, lasers and classical circuits, respectively.

7.1 Quantum system simulating PT symmetry in the LRC loop.

The LRC loop with PT symmetry was first experimentally described by Schindler et al. ^[154], proposed in 2011, consists of a pair of oscillating couplers, one with amplification and the other with equivalent attenuation.It is the gain-loss mechanism in this pair of oscillating couplers (also known as dimers) that allows this loop to break Hermiticity while maintaining PT symmetry.This "active" dimer can be realized using simple electronics, and its "phase transition" from real to complex eigenspectrum can be directly observed since it mimics a PT symmetric quantum system in a classical system.Near the phase transition point with novelty, the general mode merges, and the relative phase difference of the components is determined by the inductive coupling. The realization of this experiment has led to a series of subsequent experiments using classical systems to simulate quantum systems.2012, Lin et al. ^[155] The PT symmetric scattering is studied experimentally by applying the LRC circuit in the inductive coupling of the transmission line, and the experimental evidence for the exotic properties of the PT symmetric scattering is provided.As an easy to implement system, the LRC circuit can be used to study many other theoretical ideas. Its simplicity and accessibility to dynamic variables enable us to gain a deeper and more thorough understanding of PT symmetric scattering.2018 Choi et al. ^[92] experimentally demonstrated a circuit to simulate a general purpose APT system. They studied the steady-state and dynamic characteristics using a resistively coupled amplifying LRC resonator circuit, enabling precise parameter control and time-resolved measurements.In this experiment, they observed the phase transition point, the reverse process of PT symmetry breaking and the time evolution of energy conservation, which confirmed the unique properties of APT quantum system and provided new light wave operation technology and innovative device operation principle for the development of other fields.

7.2 PT symmetry in lasers.

Because there are not only high and low refractive index profiles but also gain and loss profiles in a semiconductor laser system, it can be used as a non-Hermitian optical system. ^[156].2007 E1-Ganainy et al. ^[157] proposed a coupled mode theory (CMT) for PT symmetric optical elements, in which each individual element and the whole system obey PT symmetry. Based on the CMT of PT symmetric optics, many related optical experiments have been produced and developed.One of the representatives is the laser with lateral and longitudinal PT symmetry, which can control its mode characteristics.2012 Miri et al. ^[158] proposes to use the lateral mode of PT symmetric structure to prepare single lateral mode laser, so as to realize the amplification of large area laser with unidirectional mode field.This PT symmetric structure can be realized by coupling two waveguides with multimode fields, one of which shows gain and the other an equal amount of loss.Researchers have used PT-symmetry-breaking transitions to achieve single-side-mode lasers, which allow the fundamental mode field to gain while maintaining the neutrality of the high-order mode field.The essential reason for the realization of this single-side-mode laser is that the coupling coefficient between the fundamental modes of the waveguide is less than that between the high-order modes, and the fundamental mode will enter the PT symmetry breaking phase earlier, thus realizing the single-side-mode laser output.Hodaei et al ^[159] realized the output of single-mode laser and the screening of longitudinal mode by using optically pumped PT symmetric ring laser with lateral double-ring coupling in 2014.2018 Yao et al. ^[37] Mode-selective PT symmetry breaking and single-mode operation are achieved by fixing the current in one waveguide and adjusting the current in the other waveguide to achieve a fixed gain in one waveguide and a variable loss in the other waveguide.

Feng et al In 2014, ^[160] proposed a laser with a single microring longitudinal PT symmetric structure using pure gain and loss modulation, which can use PT symmetric phase transition to realize the selection of inherent single-mode laser.Gu et al ^[161] In the prepared longitudinal PT symmetric stripe laser, a spectrum with a larger mode spacing is found. This is due to the fact that the laser has an asymmetric pumping region and non-pumping region, and the mode spacing increases when it enters the PT symmetric breaking phase.In addition to the mode-field separation control laser using PT symmetry, the invisibility of light at the PT symmetry phase transition point can also be used to realize the orbital angular momentum microcavity laser. ^[162,163].Because the reflectivity of the light at the longitudinal PT symmetric complex index grating is zero, only the beam output in one direction can be retained on the microring, and the unique asymmetric structure exhibited by the unidirectional destructive interference can also be used to design wave propagation and unidirectional laser emission independent of the incident direction. ^[164,165].

7.3 Classical Circuit Simulation of Non-Hermitian System

Compared with the traditional quantum platform, the classical circuit system has become a powerful platform for simulating quantum States of matter because of its unrestricted network form and high degree of freedom of regulation, which in principle can simulate quantum tight-binding models with arbitrary dimensions, arbitrary intersite transitions, and arbitrary boundary conditions.Researchers have successfully simulated an important non-Hermitian quantum model using classical circuits via the Simulator of Emulated Circuits (SPICE) The steady-state properties of the nonreciprocal Aubry-Andr André model (AA model), including the complex energy spectrum and the winding number of the energy spectrum, which embody the non-Hermitian topological properties of the system under periodic boundary conditions, and the competition between the non-Hermitian skin effect and the quasi-disordered localization under open boundaries.The researchers introduced in detail how to establish the mapping between the Laplacian form of classical circuit and the Hamiltonian matrix of quantum tight-binding model under different boundary conditions, and gave the circuit design scheme of non-reciprocal AA model under different boundary conditions.Due to the universality of the scheme, the design principles and theory discussed in this work can be directly applied to the circuit simulation of other non-Hermitian quantum models.

8. Summary and Prospect

Quantum simulation has been an important driving force and main research direction of quantum information research, and its theoretical and experimental research has developed rapidly.Non-Hermitian system, as an extension of traditional quantum mechanics, has been develope rapidly in that past two decades and has become a research hotspot because of its close relationship with open and dissipative systems and its potential application value.In this paper, we focus on the combination of the two, and briefly review the new progress of quantum simulation of non-Hermitian systems, focusing on the theoretical and experimental studies of quantum simulation of non-Hermitian systems based on LCU, including PT-symmetric and anti-symmetric systems, PT-arbitrary phase-symmetric systems, P-pseudo-Hermitian symmetric and P-pseudo-Hermitian anti-symmetric systems, pseudo-Hermitian arbitrary phase-symmetric systems. Quantum simulation of τ-antipseudohermitian symmetric systems, etc.At the same time, other research methods of quantum simulation of non-Hermitian systems are briefly introduced, including random walk, embedded and space extension.On the experimental side, several typical examples of non-Hermitian quantum simulation based on nuclear magnetic resonance, quantum optics and photonics are introduced, as well as the experimental study of non-Hermitian system simulation using classical physical systems.

At present, the theoretical research on quantum simulation of non-Hermitian systems focuses on PT symmetry and antisymmetry, and the theoretical research on other non-Hermitian systems, especially pseudo-Hermitian systems, is gradually increasing.Experiments mainly focus on PT symmetric and antisymmetric systems, and quantum simulation experiments for other non-Hermitian systems, including pseudo-Hermitian systems, are less studied, which is the development direction of non-Hermitian experiments in the future.However, non-Hermitian systems are not limited to Hamiltonians with PT symmetry or pseudo-Hermiticity, and there are more non-Hermitian Hamiltonians with different forms to be discovered, studied and applied.On the other hand, most of the non-Hermitian quantum simulation studies focus on two-level systems, mainly to show the novel properties of non-Hermitian systems. With the deepening of research, the mathematical form of high-dimensional non-Hermitian Hamiltonians and efficient quantum simulation need to be further studied.