Directed graph centrality sequencing method based on Pseudo Hermite continuous time quantum walk

Abstract

A directed graph centrality sorting method based on Pseudo Hermite continuous time quantum walk comprises the following steps: 1) a graph Pseudo Hermite detection step: judging whether an original graph meets Pseudo Hermite properties, if so, continuing to execute the method, and if not, ending the method; 2) a graph structure expansion step: adding a node i with out-degree of 0 and in-degree of N on G to form an expansion graph # imgabs0 # 3) a Hamiltonian encoding step: initializing a quantum system and setting the Hamiltonian of the system as the Hamiltonian # imgabs2 # corresponding to the expansion graph # imgabs1 # 4) an initial state preparation step: initializing the state of the quantum system as igt; a quantum state; the initial state preparation step comprises initializing the state of the quantum system into igt; and quantum states. And 5) in the centrality calculation step, calculating the average probability distribution of each node of the expansion graph # imgabs3 # by using Pseudo Hermite continuous time quantum walk, and normalizing the probability values of all nodes except the node i to obtain the centrality sequence of each node of the graph G.

Description

Technical Field

[0001] The invention relates to the fields of graph theory and quantum algorithm, and uses pseudo-Hermitian continuous-time quantum walk to realize the centrality sorting of directed graphs. Background Art

[0002] Vertex centrality is a basic concept in graph theory, which is used to measure the importance or influence of a node in a graph. In a graph composed of vertices (nodes) and edges (connections), centrality indicators help identify nodes that play a key role in the network structure or function.

[0003] There are many types of node centrality, such as degree centrality, closeness centrality, eigenvector centrality, PageRank, etc., each of which emphasizes different aspects of node importance. Degree centrality measures the number of direct connections a node has, indicating its potential influence or connectivity. Nodes with high centrality are directly connected to many other nodes and can therefore have significant influence. On the other hand, betweenness centrality evaluates how often a node appears in the shortest path between other nodes. Nodes with high betweenness centrality act as key bridges or connectors in the graph, facilitating communication or control between different parts of the network. Closeness centrality measures how close a node is to all other nodes in the graph, and is usually calculated as the inverse of the sum of the shortest path distances from the node to all other nodes. Nodes with high closeness centrality can interact with other nodes efficiently, making them essential in rapid communication or influence. Eigenvector centrality evaluates the importance of a node based on the importance of its neighboring nodes, giving importance to nodes that are connected to other highly ranked nodes. This method is very useful for detecting key nodes in a network.

[0004] PageRank It is an algorithm for ranking web pages. The algorithm is based on the link relationship between web pages and calculates the importance and ranking of web pages. Search Engines It helps users find relevant web page information. The core idea of ​​PageRank is that the importance of a web page is determined by the number and quality of references to it by other web pages. If a web page is cited by many other web pages, and the sources of these references are also important, then the importance of this web page is higher. PageRank is a variant of eigenvector centrality, originally developed by Google. It takes into account both the number and quality of inbound links and provides a more refined measure of node importance.

[0005] Node centrality is crucial in network analysis, providing insights into the structure, dynamics, and fragility of complex systems such as social, ecological, or communication systems. Understanding which nodes are central can reveal key influencers, such as opinion leaders or connectors in social networks, and provide guidance for enhancing network robustness by protecting key nodes from failures or attacks. Node centrality has been widely used in various aspects of production and life, including web page ranking[1], social network analysis[2], and identifying key nodes in transportation networks[3].

[0006] etc. Centrality indicators can also achieve resource optimization, such as targeted distribution of information or efficient traffic management. In epidemic models, centrality helps identify super spreaders, thereby guiding intervention strategies such as vaccination or isolation to control the spread of the disease. In addition, centrality also plays a role in understanding community structure and network modularity, revealing the formation and interaction of communities in the network.

[0007] There are many classic centrality ranking algorithms, such as degree centrality, betweenness centrality, closeness centrality, eigenvector centrality, PageRank centrality, etc. Different centrality rankings focus on different network characteristics and are therefore suitable for different scenarios. In the field of quantum algorithms, both discrete-time quantum walk models [4, 5] and continuous-time quantum walk models [6-8] can be used to construct quantum centrality ranking algorithms for undirected graphs. However, many graphs abstracted from practical problems are directed graphs, and standard quantum walk models cannot be used to analyze the node centrality information of directed graphs. By introducing non-unitary evolution or improving quantum walk models, algorithms such as pseudo-Hermitian continuous-time quantum walks can be used to construct node centrality algorithms for directed graphs.

[0008] Compared with existing technologies, this algorithm can reveal the network centrality more deeply by utilizing the dynamic characteristics of quantum walks, thereby enhancing the ability to identify and effectively utilize key nodes. Xue Peng's research group realized non-unitary continuous-time quantum walks on a three-node graph on a bulk optical platform and demonstrated a node centrality sorting algorithm for a three-node directed graph [9]. This method of realizing non-unitary evolution requires the expansion of the state space. For continuous-time quantum walks on an N-node graph, a 2N-dimensional state space is required to achieve it. There are also experimental schemes on photonic chips to achieve non-unitary evolution by increasing the dimension of the state space

[10] . In 2017, Izaac et al. used the pseudo-Hermitian property of the Hamiltonian of a directed graph to realize a quantum centrality sorting algorithm for a directed graph based on pseudo-Hermitian continuous-time quantum walks without increasing the state space overhead and introducing decoherence

[11] . In 2020, based on this scheme, Wu Tong et al. used bulk optical components to implement a quantum centrality sorting algorithm on three-node and four-node directed graphs, and expanded the three-node directed graph to a nine-node directed graph through a two-photon experiment, demonstrating the scalability of the centrality sorting algorithm based on pseudo-Hermitian continuous-time quantum walks

[12] . However, due to the stability of the optical path and the volume of optical components, it is difficult to achieve large-scale expansion of the centrality sorting problem based on quantum walks on a bulk optical platform.

[0009] References

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[0021]

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[0022]

[11] Izaac JA, Zhan X, Bian Z, et al. Centrality measure based on continuous-time quantum walks and experimental realization[J]. Physical ReviewA.2017,95(3):032318.

[0023]

[12] Wu T,Izaac JA,Li ZX,et al.Experimental parity-time symmetricquantum walks for centrality ranking on directed graphs[J].Phys.Rev.Lett.2020,125:240501. Summary of the invention

[0024] The purpose of this invention is to propose a directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walk.

[0025] The technical solution of the present invention is: a directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walk, which includes five parts: graph pseudo-Hermitian detection, graph structure expansion, Hamiltonian encoding, initial state preparation, and centrality calculation. The workflow of the technical solution and the relative relationship between the five parts are as follows: Figure 1shown.

[0026] Assume that the original graph G is an N-node graph,

[0027] In the pseudo-Hermitian property detection step, it is determined whether the original image satisfies the pseudo-Hermitian property, if so, the method continues to be executed, otherwise the method ends;

[0028] The graph pseudo-Hermitian property detection step includes: determining whether the original graph satisfies the pseudo-Hermitian property, if so, continuing to execute the method, otherwise terminating the method, and clarifying the mapping relationship between the directed graph and the Hamiltonian as follows. Let G = (V, E) be a directed graph, where V is the set of vertices of the graph, and E is the set of directed edges of the graph.

[0029] Construct the adjacency matrix A of the directed graph G = (V, E):

[0030] And the diagonal matrix D:

[0031] D jj = deg(j), deg(j) represents the out-degree of node j. H = A + D is defined as the Hamiltonian corresponding to the directed graph G. The pseudo-Hermitian property test of the directed graph G includes three different but equivalent judgment criteria: ① H is similar to its conjugate transpose, that is, there exists a Hermitian linear automorphism operator Θ with ② Determine whether H has a biorthogonal basis (a diagonalizable matrix has a biorthogonal basis) and the eigenspectrum is real. ③ H is diagonalizable and the eigenspectrum is real.

[0032] In the graph structure expansion step, a new node i with out-degree 0 and in-degree N is added to G to form an extended graph

[0033] In the Hamiltonian encoding step, the quantum system is initialized and the system Hamiltonian is set to the expanded graph The corresponding Hamiltonian

[0034] In the initial state preparation step, the state of the quantum system is initialized to the |i> quantum state; the initial state preparation step includes initializing the state of the quantum system to the |i> quantum state: a method for preparing the initial state, in an optical system based on path coding, coupling photons to the i-th waveguide can complete the preparation process of the quantum state |i> (Example 1).

[0035] In the centrality calculation step, a pseudo-Hermitian continuous-time quantum walk is used to calculate the extended graph The average probability distribution of each node is calculated, and the probability values ​​of all nodes except the i node are normalized to obtain the centrality ranking of each node in the graph G; the centrality calculation step includes using pseudo-Hermitian continuous-time quantum walk to calculate the average probability distribution of each node in the extended graph, and normalizing the probability values ​​of all nodes except the newly added nodes to obtain the centrality ranking of each node in the original graph; a quantum algorithm for implementing centrality sorting in a directed graph obtains the influence index of the node by calculating the average probability distribution of the quantum state.

[0036] In the graph structure expansion step, a new node i with out-degree 0 and in-degree N is added to G to form an extended graph In the Hamiltonian encoding step, the quantum system is initialized and the system Hamiltonian is set to the expanded graph The corresponding Hamiltonian In the initial state preparation step, the quantum system state is initialized to the |i> quantum state. In the centrality calculation step, the pseudo-Hermitian continuous-time quantum walk is used to calculate the extended graph The average probability distribution of each node is calculated, and the probability values ​​of all nodes except the i node are normalized to obtain the centrality ranking of each node in the graph G.

[0037] Furthermore, the Hamiltonian encoding step includes initializing the quantum system and setting the system Hamiltonian to the Hamiltonian corresponding to the extended graph. A quantum system initialization method based on pseudo-Hermitian Hamiltonian, wherein the construction method of the Hamiltonian of the quantum system is dynamically set according to the structure of the extended graph. Because different directed graphs correspond to different extended graph structures, the corresponding system Hamiltonians are also different, so the Hamiltonian encoding step must be able to be dynamically set according to the specific structure of the extended graph.

[0038] In optical quantum systems, quantum state initialization, Hamiltonian encoding and evolution, and centrality calculation are achieved through photon paths and interference.

[0039] The method realizes quantum state initialization, Hamiltonian encoding and evolution, and centrality calculation in a superconducting quantum computing system through superconducting quantum bits and circuit quantum electrodynamics.

[0040] The technical solution can be run on a variety of quantum computing platforms, including but not limited to optical quantum systems, superconducting quantum computing systems, ion trap quantum computing systems and other quantum hardware platforms that support pseudo-Hermitian continuous-time quantum walks.

[0041] Beneficial effects of the present invention: Compared with the existing directed graph centrality sorting method based on quantum walk, the method of the present invention has the following advantages: In the Hamiltonian encoding and ground state preparation steps, the present invention only needs to introduce a one-dimensional additional Hilbert space, which greatly saves space costs. The present invention only needs to prepare the ground state, and does not need to prepare the complete superposition state, which simplifies the preparation process of the quantum state. The present invention has good scalability and can be applied to multi-photon situations.

[0042] The above advantages make the designed directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walk easy to implement experimentally. In the specific implementation method, we will give the experimental results under different abstract graph models. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a flow chart of the directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk of the present invention;

[0044] Figure 2 It is a quantum optical system implementation scheme for the directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk;

[0045] Figure 3 A and B are the three-node graph given in Example 2 and the theoretical and experimental results of the centrality of each corresponding node; they are the theoretical values ​​of the centrality of the Erdos-Renyi graph given for the randomly generated Erdos-Renyi graph and the directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk.

[0046] Figure 4 A and B are the theoretical results of the 15-node graph and the corresponding node centrality given in Example 3. are the theoretical and experimental values ​​of the centrality of a 3-node directed graph given for a given 3-node directed graph and the directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walk. DETAILED DESCRIPTION

[0047] The following is a detailed introduction to the details of each part.

[0048] 1. Pseudo-Hermitian detection

[0049] Let G = (V, E) be a directed graph, where V is the set of vertices in the graph and E is the set of directed edges in the graph. Construct the adjacency matrix A of the directed graph G = (V, E):

[0050]

[0051] D jj= deg(j), deg(j) represents the out-degree of node j. H = A + D is defined as the Hamiltonian corresponding to the directed graph G. Where A and D are the adjacency matrix and diagonal matrix of the directed graph G defined above, respectively.

[0052] make is the inner product space corresponding to the vertex set V (i.e., the standard basis state of the space corresponding to the vertex), and the linear operator If there exists a Hermitian linear automorphism operator Θ with Then the operator O is called pseudo-Hermitian. A non-Hermitian Hamiltonian is pseudo-Hermitian if it has a biorthogonal basis and the eigenspectrum is real.

[0053] This gives three criteria for detecting pseudo-Hermitian properties of graphs:

[0054] ④ Judge that H is similar to its conjugate transpose, that is, there exists a Hermitian linear automorphism operator Θ with

[0055] ⑤ Determine whether H has a biorthogonal basis (a diagonalizable matrix has a biorthogonal basis) and the eigenspectrum is real.

[0056] ⑥H can be diagonalized and the eigenspectrum is real.

[0057] There are actually three ways to introduce non-Hermitian matrices. The first is to introduce non-diagonal Non-reciprocity The second is to introduce the gain and loss of reciprocal transitions; the third is to introduce Virtual mass , that is, the gain and loss of the diagonal items. The introduction of these three methods will bring about differences in symmetry.

[0058] 2. Graph Structure Extension

[0059] Assume that graph G is an N-node graph. Add a node i with out-degree 0 and in-degree N to G (for the convenience of calculation, the i node is usually set to 0 node) to form an extended graph Get the extended graph The corresponding Hamiltonian is:

[0060]

[0061] 3. Hamiltonian encoding

[0062] Further correcting the Hamiltonian, we get:

[0063]

[0064] η can be defined using a biorthogonal basis matrix. Assume E n and They are The nth eigenvalue and eigenvector of are:

[0065]

[0066] Another set of bases for H can be constructed satisfy:

[0067]

[0068] but:

[0069]

[0070] 4. Initial preparation

[0071] Select initial state That is, the quantum state corresponding to the newly added node i is selected as the initial state. For the convenience of calculation, the i node is usually set to the 0 node, that is,

[0072] 5. Centrality calculation

[0073] After a period of time t, the final state is obtained:

[0074]

[0075] Calculate the average probability of node w:

[0076]

[0077] Normalize the probability values ​​of all nodes except node i to get the centrality ranking of each node in graph G, that is, the centrality of node j in graph G is:

[0078]

[0079] Example 1: In Figure 2 The system shown in the figure has completed the implementation of the directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk. The system mainly consists of two parts: on-chip and off-chip. The on-chip part includes the light source and MZI network, and the off-chip part mainly includes superconducting single-photon detectors and coincidence counting.

[0080] The on-chip light source includes a predicted single photon source and a DWDM multi-photon (channel) light source. The micro-ring resonant cavity coupled with the unequal-arm interferometer is adjusted to align the resonance wavelengths of the micro-ring pump, signal and idle light with the 33, 29 and 37 channels of the DWDM respectively (the DWDM dense wavelength division multiplexing system mainly consists of the following parts:

[0081] Wavelength multiplexer: combines multiple optical signals of different wavelengths together. It is usually composed of multiple wavelength selectors. Through optical elements such as gratings and fiber Bragg gratings, optical signals of different wavelengths are arranged in a predetermined order to form a multi-wavelength optical signal. Optical amplifier: such as erbium-doped fiber amplifier (EDFA) and Raman fiber amplifier (RFA), it is used to enhance optical signals and ensure the quality and stability of signals during long-distance transmission. Wavelength demultiplexer: In contrast to the wavelength multiplexer, it separates multi-wavelength optical signals in a predetermined order and restores them to the original single-wavelength optical signals. Optical receiver: converts the received optical signal into an electrical signal. Since the optical signal transmitted from the optical fiber is generally very weak, it has higher requirements for the optical detector. Optical monitoring channel: used to carry the management and monitoring of the DWDM system, so that the network management system can effectively manage the DWDM system. Transmitting and receiving active parts: including optical transmitters and optical receivers, responsible for sending and receiving signals. Passive part of combining and splitting: the combining and splitting of optical signals are completed through combiners and splitters. Optical transmission and optical amplification part: including optical fiber, optical amplifier, etc., responsible for signal transmission and amplification.

[0082] These parts work together to achieve the multi-wavelength transmission and high-bandwidth capabilities of the DWDM system.

[0083] A continuous laser with a central wavelength of 1550.9nm and a power of 4mw is coupled into the chip as pump light after the sideband noise is filtered out by the 33-channel DWDM. The microring is operated in an over-coupled state by adjusting the long arm phase of the microring unequal-arm interferometer. For the prediction single-photon source, two unequal-arm interferometers are connected to the output end of the microring photon source. The first unequal-arm interferometer filters out the pump light transmitted from the microring, and the second unequal-arm interferometer outputs the signal photon and the idle photon to different paths respectively. The idle photon is used as a prediction photon, directly coupled out of the chip through the grating, and connected to the superconducting nanowire single-photon detector after being filtered by the 37-channel DWDM. The signal photon enters the square network composed of Mach-Zehnder interferometers for pseudo-Hermitian continuous-time quantum walk. For the multi-photon source, only one unequal-arm interferometer is connected to the output end of the microring photon source to filter out the pump light transmitted from the microring. Both the signal and idle photons enter the square network composed of Mach-Zehnder interferometers for pseudo-Hermitian continuous-time quantum walk.

[0084] The on-chip MZI network is a 7-dimensional universal linear optical network that uses path encoding to encode |1>,|2>,…,|7> from top to bottom.

[0085] To complete the centrality sorting of a directed graph based on pseudo-Hermitian continuous-time quantum walk, it is necessary to complete the four steps of initial state preparation, quantum walk, photon detection, and data processing in sequence.

[0086] (1) Initial state preparation: This implementation uses a path encoding scheme to encode the system ground state. There are a total of 7 waveguides, and a single photon in the kth waveguide represents the ground state |k>. Therefore, in order to complete the centrality sorting of the directed graph based on pseudo-Hermitian continuous-time quantum walk, we only need to couple the photon from the i-th waveguide into the chip to complete the preparation of the initial state |i>.

[0087] (2) Quantum walk: According to the extended graph Hamiltonian, modified Hamiltonian Get the unitary evolution operator We need to configure the interferometer network as U(t), where t is the quantum walk time. The construction of the interferometer network is based on the Reck decomposition principle. The structure of an MZI plus an external phase shifter can be represented by a 2*2 matrix:

[0088]

[0089] Where ω represents the phase of the internal phase shifter of MZI, and φ represents the phase of the external phase shifter of MZI. Construct an N*N matrix T m,n , the matrix is ​​an N*N identity matrix, where I mm ,I mn ,I nm ,I nn The elements are respectively replaced by T in the T matrix 11 ,T 12 ,T 21 ,T 22 An arbitrary N*N unitary matrix U(N) and a series of ordered T m,n Matrix multiplication has the following relationship:

[0090]

[0091] And so on:

[0092] U(N)·T N,N-1 ·T N,N-2 …T 2,1 =D

[0093] Where D is a diagonal matrix, which is expressed as a global phase in optical experiments and does not affect the experimental results. It can be obtained that any N*N unitary matrix U(N) can be represented by a series of MZI plus external phase shifter structures. The MZI network constructed based on this scheme is a triangular network in topology. By changing the arrangement of T m,nThe matrix order can transform a triangular network into a rectangular network. Therefore, the MZI network constructed based on this scheme can obtain U(t) by configuring the corresponding phase shifter. U(t) is applied to the initial state of the system prepared in the previous step, and the photons complete the Hamiltonian preparation and system evolution process. Then, the photons are coupled from the chip to perform photon pair detection.

[0094] (3) Photon pair detection. After the photons that complete the pseudo-Hermitian continuous-time quantum walk in the universal linear optical network are coupled out of the chip, they are filtered through the 37-channel and 29-channel DWDM channels and then connected to the superconducting nanowire single-photon detector. By optimizing the polarization controller before each channel of the detector, the maximum photon count of each channel is obtained. The outputs of all detection channels are connected to the coincidence counting module to measure the coincidence counts of signal photons and idle photons within 3 seconds. After deducting the losses of each path, the probability of the pseudo-Hermitian continuous-time quantum walk corresponding to each node on the directed graph is obtained by normalizing the coincidence counts of signal photons and idle photons.

[0095] (4) Calculation of the centrality of each node. The probability of each node detected is averaged over all simulation times, and the average values ​​of all nodes except node i are normalized to obtain the centrality ranking of each node in graph G.

[0096] Example 2

[0097] Figure 3 3A (left figure) is a given 3-node directed graph. This directed graph is generated by mathematical modeling of a small-scale social network, where each vertex corresponds to a person, and the directed edge corresponds to the acquaintance relationship between two people. For example, vertex A points to vertex B, which means that A knows B (but B knows A). The following is the workflow of the directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk in the optical system experimental operation built in Example 1 and the simulation operation on a classical computer:

[0098] (1) Give the Hamiltonian H corresponding to the graph:

[0099]

[0100] The Hamiltonian is diagonalizable and its eigenspectrum is real, satisfying the pseudo-Hermitian property.

[0101] (2) Obtain the extended graph Hamiltonian

[0102]

[0103] (3) Select a biorthogonal basis:

[0104]

[0105] calculate have to:

[0106]

[0107]

[0108] (4) Select the initial state

[0109] (5) After calculating the centrality of each node, Figure 3 As shown in the right figure, the solid-line frame represents the theoretical value of the centrality ranking obtained by simulation on a computer, and the dotted-line frame represents the experimental value of the centrality ranking obtained by experimentally running the optical system built in Example 1.

[0110] The results of Example 2 include both the experimental results of the directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk on the chip of Example 1, and the simulation results of running the same method on a computer.

[0111] Example 3

[0112] Figure 4 The above figure is a randomly generated Erdos-Renyi graph. This directed graph is generated by mathematical modeling of a larger social network. Each vertex corresponds to a person, and the directed edge corresponds to the acquaintance relationship between two people. For example, vertex A points to vertex B, which means that A knows B (but B knows A). Theoretical research shows that the Erdos-Renyi graph is a very effective mathematical modeling method for large-scale social networks. Use the python software package NetworkX to generate a directed Erdos-Renyi graph, where N=15 and P=0.3, and then select a graph that satisfies the pseudo-Hermite property. The directed graph centrality sorting method based on pseudo-Hermite continuous-time quantum walk is simulated on a classical computer, and the centrality of each node in the generated Erdos-Renyi graph is calculated as follows Figure 4 The invented method sorts the centrality of the Erdos-Renyi graph, and the person with the highest centrality is the most important person in the social network corresponding to the Erdos-Renyi graph (ie, the person has the most people who know him).

[0113] NetworkX can only generate directed graphs, but cannot give a centrality ranking of directed graphs. There are some classical algorithms that can give a centrality ranking, but the quantum method designed by the present invention can give a new centrality ranking method that is different from the existing classical methods, and has a potential advantage in speed: because of the pseudo-Hermitian continuous time quantum walk, because this method uses quantum effects, multiple photons can be searched simultaneously, so in theory there will be a speed advantage.

[0114] On the other hand, the centrality ranking given by the present invention may have an advantage in accuracy. This is because the concept of "centrality" itself is a relative concept and there is no absolute value. For example, which person is the most important in a social network is obviously a subjective concept. The method of the present invention can provide a new ranking method, which can be used together with the classical method to provide more reference information for social networks or other practical applications that require the use of centrality information.

[0115] The results of Example 3 only include the simulation results of running the directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk on a computer.

[0116] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walk, characterized by: It includes five steps: graph pseudo-Hermitian detection, graph structure expansion, Hamiltonian encoding, initial state preparation, and centrality calculation. Assuming that the original graph G is an N-node graph, 1) Graph pseudo-Hermitian property detection step, determine whether the original graph satisfies the pseudo-Hermitian property, if so, continue to execute the method, otherwise end the method; clarify the mapping relationship between the directed graph and the Hamiltonian as follows: Let G = (V, E) be a directed graph, where V is the set of vertices of the graph and E is the set of directed edges of the graph. The adjacency matrix A of the directed graph G = (V, E) is defined as: The diagonal matrix D of the directed graph G = (V, E) is defined as: D jj = deg(j), deg(j) represents the out-degree of node j; define H = A + D as the Hamiltonian corresponding to the directed graph G, where A and D are the adjacency matrix and diagonal matrix of the directed graph G defined above. The pseudo-Hermitian property test of the directed graph G contains three different but equivalent judgment criteria: ① H is similar to its conjugate transpose, that is, there exists a Hermitian linear automorphism operator Θ with ② Determine whether H has a biorthogonal basis (a diagonalizable matrix has a biorthogonal basis) and the eigenspectrum is real; ③ H is diagonalizable and the eigenspectrum is real. 2) Graph structure expansion step: add a node i with out-degree 0 and in-degree N to G to form an extended graph 3) Hamiltonian encoding step, initializing the quantum system and setting the system Hamiltonian to the extended graph The corresponding Hamiltonian 4) Initial state preparation step, initializing the state of the quantum system to the quantum state |i>; The initial state preparation step includes initializing the state of the quantum system to the quantum state |i>. In the optical system based on path coding, coupling the photon to the i-th waveguide can complete the preparation process of the quantum state |i>; 5) In the centrality calculation step, pseudo-Hermitian continuous-time quantum walk is used to calculate the extended graph The average probability distribution of each node, and normalize the probability values ​​of all nodes except the i node, to get the centrality ranking of each node in the graph G; The centrality calculation steps include: using pseudo-Hermitian continuous-time quantum walk to calculate the average probability distribution of each node in the expanded graph, and normalizing the probability values ​​of all nodes except the newly added nodes to obtain the centrality ranking of each node in the original graph; implementing a quantum algorithm for centrality sorting in a directed graph, which obtains the influence index of the node by calculating the average probability distribution of the quantum state.

2. The directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk according to claim 1, characterized in that: The Hamiltonian encoding step includes initializing the quantum system and setting the system Hamiltonian to the Hamiltonian corresponding to the extended graph; a quantum system initialization method based on the pseudo-Hermitian Hamiltonian, in which the construction method of the Hamiltonian of the quantum system is dynamically set according to the structure of the extended graph.

3. The directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walk according to claim 1 is characterized in that: Pseudo-Hermitian detection step, let G = (V, E) be a directed graph, where V is the set of vertices of the graph, E is the set of directed edges of the graph, and construct the adjacency matrix A of the directed graph G = (V, E): And the diagonal matrix D:D jj = deg(j), deg(j) represents the out-degree of node j, and H = A + D is defined as the Hamiltonian corresponding to the directed graph G; make is the inner product space corresponding to the vertex set V (i.e., the standard basis state of the space corresponding to the vertex), and the linear operator If there exists a Hermitian linear automorphism operator Θ with Then the operator 0 is called pseudo-Hermitian; a non-Hermitian Hamiltonian has a biorthogonal basis and the eigenspectrum is real, then the Hamiltonian is pseudo-Hermitian; This gives three criteria for detecting pseudo-Hermitian properties of graphs: ① Judge that H is similar to its conjugate transpose, that is, there exists a Hermitian linear automorphism operator Θ with ② Determine that H has a biorthogonal basis (a diagonalizable matrix has a biorthogonal basis) and the eigenspectrum is real; ③H can be diagonalized and the eigenspectrum is real.

4. The directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk according to claim 1, characterized in that: In the graph structure expansion step, assume that graph G is an N-node graph, and add a node i with out-degree 0 and in-degree N to G to form an expanded graph Get the extended graph The corresponding Hamiltonian is:

5. The directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk according to claim 1, characterized in that: The Hamiltonian is further corrected in the Hamiltonian encoding step to obtain: η can be defined using a biorthogonal basis matrix; assuming that En and They are The nth eigenvalue and eigenvector of are: Construct another basis of H |φ> that satisfies: but:

6. The directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walk according to claim 1, characterized in that Initial state preparation, select initial state That is, the quantum state corresponding to the newly added node i is selected as the initial state; for the convenience of calculation, the i node is usually set to 0 node, that is, Centrality calculation: After a period of time t, the final state is obtained: Calculate the average probability of node w: Normalize the probability values ​​of all nodes except node i to get the centrality ranking of each node in graph G, that is, the centrality of node j in graph G is:

7. The directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk according to claim 1, characterized in that: The method can be run on a variety of quantum computing platforms, including but not limited to optical quantum systems, superconducting quantum computing systems, ion trap quantum computing systems and other quantum hardware platforms that support pseudo-Hermitian continuous-time quantum walks.

8. The directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk according to claim 1, characterized in that: In optical quantum systems, quantum state initialization, Hamiltonian encoding and evolution, and centrality calculation are achieved through photon paths and interference.

9. The directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk according to claim 1, characterized in that: The method realizes quantum state initialization, Hamiltonian encoding and evolution, and centrality calculation in a superconducting quantum computing system through superconducting quantum bits and circuit quantum electrodynamics.

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