Directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walk

Abstract

A directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walks includes: 1) a graph pseudo-Hermitian property detection step, which determines whether the original graph satisfies the pseudo-Hermitian property. If so, the method continues execution; otherwise, the method terminates; 2) a graph structure expansion step, which adds a node i with an out-degree of 0 and an in-degree of N to G to form an extended graph; 3) a Hamiltonian encoding step, which initializes the quantum system and sets the system Hamiltonian to the Hamiltonian corresponding to the extended graph; 4) an initial state preparation step, which initializes the quantum system state to the |i> quantum state; the initial state preparation step includes initializing the quantum system state to the |i> quantum state. 5) In the centrality calculation step, a pseudo-Hermitian continuous-time quantum walk is used to calculate the average probability distribution of each node in the extended graph, and the probability values ​​of all nodes except node i are normalized to obtain the centrality ranking of each node in the graph G.

Description

Technical Field

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

[0002] Vertex centrality is a fundamental concept in graph theory that measures the importance or influence of a node in a graph. In a graph consisting of vertices (nodes) and edges (connections), centrality metrics help identify nodes that play a key role in the network's structure or function.

[0003] There are several types of node centrality, such as degree centrality, closeness centrality, eigenvector centrality, and PageRank, each emphasizing 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 betweenness centrality are directly connected to many other nodes and thus have the potential to exert significant influence. On the other hand, betweenness centrality assesses how frequently a node appears in the shortest paths between other nodes. Nodes with high betweenness centrality act as key bridges or connectors in a graph, facilitating communication or control between different parts of the network. Closeness centrality measures a node's proximity to all other nodes in the graph and is typically 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 efficiently with other nodes, making them crucial for rapid communication or influence. Eigenvector centrality assesses a node's importance based on the importance of its neighbors, prioritizing nodes connected to other highly ranked nodes. This approach is 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 Engine PageRank helps users find relevant web information. The core idea behind PageRank is that the importance of a webpage is determined by the number and quality of its citations from other webpages. The more a webpage is cited by other webpages, and the sources of these citations are also important, the more important it is. PageRank is a variation of eigenvector centrality, originally developed by Google, that considers both the quantity and quality of inbound links, providing a more refined measure of node importance.

[0005] Node centrality is crucial in network analysis, providing insights into the structure, dynamics, and vulnerability of complex systems such as social, ecological, or communication systems. Understanding which nodes are central nodes 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] Centrality metrics can also optimize resources, such as targeted information distribution or efficient traffic management. In epidemic models, centrality helps identify superspreaders, thereby guiding intervention strategies such as vaccination or quarantine to control disease spread. Furthermore, centrality plays a role in understanding community structure and network modularity, revealing how communities form and interact within a network.

[0007] Currently, there are many classic centrality ranking algorithms, including degree centrality, betweenness centrality, closeness centrality, eigenvector centrality, and PageRank centrality. Different centrality ranking algorithms focus on different network characteristics, making them 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 node centrality information in directed graphs. By introducing nonunitary 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 ranking 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. There are also experimental schemes on photonic chips that realize non-unitary evolution by increasing the state space dimension

[10] . In 2017, Izaac et al. used the pseudo-Hermitian property of the Hamiltonian of a directed graph to realize a quantum centrality ranking algorithm for a directed graph based on pseudo-Hermitian continuous-time quantum walks without increasing the state space overhead and without 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 limitations of the optical path stability 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, the method continues to execute; otherwise, the method terminates. The mapping relationship between the directed graph and the Hamiltonian is as follows. Let G = (V, E) be a directed graph, where V is the set of vertices and E is the set of directed edges.

[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 its eigenspectrum is real. ③ H is diagonalizable and its 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 quantum system state is initialized to the |i> quantum state; the initial state preparation step includes initializing the quantum system state 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, 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 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; 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 node i are normalized to obtain the centrality ranking of each node in graph G.

[0037] Furthermore, the Hamiltonian encoding step involves initializing the quantum system and setting the system Hamiltonian to the Hamiltonian corresponding to the extended graph. This is a quantum system initialization method based on the pseudo-Hermitian Hamiltonian, in which the Hamiltonian of the quantum system is constructed dynamically based on the structure of the extended graph. Because different directed graphs correspond to different extended graph structures, and therefore their corresponding system Hamiltonians, the Hamiltonian encoding step must be able to dynamically set the Hamiltonian based on 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] The present invention has the following advantages over existing methods for sorting directed graph centrality based on quantum walks: During the Hamiltonian encoding and ground state preparation steps, the present invention only requires the introduction of a one-dimensional additional Hilbert space, significantly saving space costs. The present invention only requires the preparation of the ground state, eliminating the need for a full superposition state, simplifying the quantum state preparation process. The present invention also exhibits excellent scalability and is applicable to multi-photon scenarios.

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

[0043] Figure 1 This 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 walks;

[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, respectively.

[0046] Figure 4 A and B are the theoretical results of the centrality of the 15-node graph and the corresponding nodes given in Example 3. They are the theoretical and experimental values ​​of the centrality of the 3-node directed graph given for the given 3-node directed graph and the directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walks, respectively. 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 and E is the set of directed edges. 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. 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, respectively.

[0052] make is the inner product space corresponding to the vertex set V (i.e., the standard basis state of the vertex corresponding to the space), 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 a real eigenspectrum.

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

[0054] ④ Determine 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 Nonreciprocity The second is to introduce the gain and loss of reciprocal transitions; the third is to introduce Virtual mass , that is, the diagonal gain and loss are used to achieve this. The introduction of these three methods will bring about different symmetries.

[0058] 2. Graph Structure Extension

[0059] Assume that the 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 expanded 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. In order to facilitate calculation, the i node is usually set to 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 here implements a directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walks. The system consists of two main components: an on-chip light source and MZI network, and an off-chip superconducting single-photon detector and coincidence counter.

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

[0081] A wavelength multiplexer combines multiple optical signals of different wavelengths. It typically consists of multiple wavelength selectors. Using optical components such as gratings and fiber Bragg gratings, it arranges the different wavelengths of optical signals in a predetermined order, creating a multi-wavelength optical signal. Optical amplifiers, such as erbium-doped fiber amplifiers (EDFAs) and Raman fiber amplifiers (RFAs), are used to amplify optical signals and ensure signal quality and stability during long-distance transmission. A wavelength demultiplexer, in contrast to a wavelength multiplexer, separates multi-wavelength optical signals according to a predetermined order, restoring them to their original single-wavelength signals. An optical receiver converts received optical signals into electrical signals. Because optical signals transmitted through optical fibers are typically very weak, high requirements are placed on the optical detector. An optical supervisory channel (OMC) carries out management and monitoring of the DWDM system, enabling network management systems to effectively manage the system. The active transmitting and receiving components include optical transmitters and receivers, responsible for signal transmission and reception. The passive combining and demultiplexing components combine and demultiplex optical signals using combiners and demultiplexers. 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 sideband noise is filtered out by the 33-channel DWDM. By adjusting the phase of the long arm of the microring unequal-arm interferometer, the microring is operated in an overcoupled state. For the prediction single-photon source, two unequal-arm interferometers are connected to the output of the microring photon source. The first unequal-arm interferometer filters out the pump light transmitted from the microring, while the second unequal-arm interferometer outputs the signal photon and idle photon to different paths. The idle photon, as the prediction photon, is directly coupled out of the chip through the grating. After being filtered by the 37-channel DWDM, it is connected to the superconducting nanowire single-photon detector. The signal photon enters a square network composed of Mach-Zehnder interferometers to perform a 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 seven waveguides in total, and a single photon in the kth waveguide represents the ground state |k>. Therefore, to complete the directed graph centrality sorting based on pseudo-Hermitian continuous-time quantum walks, 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 T matrix 11 ,T 12 ,T 21 ,T 22 An arbitrary N*N unitary matrix U(N) and a series of sequentially arranged 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 sequence can transform a triangular network into a rectangular one. Therefore, the MZI network constructed based on this scheme can be configured with the appropriate phase shifters to obtain U(t). Applying U(t) to the initial state of the system prepared in the previous step, photons complete the Hamiltonian preparation and system evolution process. The photons are then coupled from the chip for 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 respectively 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. All detection channel outputs are connected to the coincidence counting module to measure the coincidence count 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 count of signal photons and idle photons.

[0095] (4) Calculation of the centrality of each node. The probability of each node obtained by detection 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) is a given three-node directed graph. This directed graph is generated by mathematically modeling a small-scale social network. Each vertex corresponds to a person, and the directed edges correspond to the acquaintance relationship between two people. For example, vertex A pointing to vertex B means that A knows B (but B also knows A). The following is the workflow of the directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walks, run in the optical system experimentally constructed in Example 1, and simulated 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 box represents the theoretical value of the centrality ranking obtained by simulation on a computer, and the dotted-line box represents the experimental value of the centrality ranking obtained by experimental running of the optical system built in Example 1.

[0110] The results of Example 2 include both experimental results of the directed graph centrality sorting method based on pseudo-Hermitian continuous-time quantum walk on the chip of Example 1, and 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-scale 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 studies have shown 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-Hermitian 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 (i.e., the person has the most people who know him).

[0113] NetworkX can only generate directed graphs, but it cannot provide a centrality ranking for them. While some classical algorithms can provide centrality ranking, the quantum method designed in this paper offers a new and different centrality ranking method than existing classical methods, with potential speed advantages. Because this method utilizes quantum effects and can search with multiple photons simultaneously, it theoretically offers a speed advantage.

[0114] On the other hand, the centrality ranking proposed by this invention may offer advantages in accuracy. This is because the concept of "centrality" itself is a relative one, without an absolute value. For example, determining who is the most important person in a social network is clearly a subjective concept. The method proposed by this invention provides a new ranking method that can be used alongside traditional methods, providing more reference information for social networks or other practical applications that require centrality information.

[0115] The results of Example 3 only include the simulation results of running the directed graph centrality ranking 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 principles 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 walks, 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 executing the method; otherwise, terminate the method. The mapping relationship between the directed graph and the Hamiltonian is as follows: Let G = (V, E) be a directed graph, where V is the set of vertices and E is the set of directed edges. 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, respectively; 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 its eigenspectrum is real; ③ H is diagonalizable and its eigenspectrum is real; 2) Graph structure expansion step: add a new node i with out-degree 0 and in-degree N to G to form an extended graph 3) Hamiltonian encoding step, initialize the quantum system and set the system Hamiltonian to the expanded graph The corresponding Hamiltonian 4) Initial state preparation step, initializing the quantum system state to the quantum state |i>; the initial state preparation step includes initializing the quantum system state to the quantum state |i>; in the path-encoded optical system, coupling a 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 node i, to obtain the centrality ranking of each node in graph G; The centrality calculation steps include: using pseudo-Hermitian continuous-time quantum walks 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 node influence index by calculating the average probability distribution of quantum states.

2. The directed graph centrality ranking 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 and E is the set of directed edges of the graph. 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 vertex corresponding to the space), 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 us three criteria for detecting pseudo-Hermitian properties of graphs: ① Determine that 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; ③H can be diagonalized and the eigenspectrum is real.

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

5. The directed graph centrality ranking method based on pseudo-Hermitian continuous-time quantum walk according to claim 1, characterized in that: The Hamiltonian is further modified 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 ranking 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 ranking 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 ranking 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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