Quantum-classical cooperative wireless link scheduling optimization method based on divide-and-conquer QUBO model and anti-non-adiabatic driving

Abstract

The invention discloses a quantum-classical collaborative wireless link scheduling optimization method based on a divide-and-conquer QUBO model and inverse non-adiabatic driving, and belongs to the technical field of quantum computing and wireless communication. According to the method, link interference in wireless communication is modeled into a conflict graph, namely, vertexes represent links, interference represents edges, utility functions on the links represent weights of the vertexes (numbers on the left of the vertexes are numbers of the vertexes, and numbers on the right of the vertexes are weights of the vertexes), and optimal wireless link scheduling is a weighted maximum independent set (WMIS) on the conflict graph. According to the method, a divide-and-conquer QUBO model is provided, the whole space is divided into Dicke state subspaces with different Hamming weights, XY-Mixer is adopted as a hybrid operator, and evolution of QAOA in the subspaces is limited. On the basis, a DQAOA algorithm is provided; a greedy algorithm and an expanded Lov-alpha-sz positive semidefinite plan are called to obtain a lower bound and an upper bound of a WMIS solution respectively; an anti-non-adiabatic driving item is introduced to modify a QAOA evolution formula, an XY-Mixer angatz is matched to enable the algorithm to be subjected to parallel evolution in a subspace, convergence is accelerated, and an optimal wireless link scheduling scheme is obtained.

Description

Technical Field

[0001] The present invention belongs to the field of quantum computing and wireless communication technology, and is a QAOA variant algorithm used to find an optimal wireless link scheduling solution. Background Art

[0002] Quantum computing is a cutting-edge technology in the field of computer science. Compared with traditional classical computing, quantum computing has the characteristics of parallelism and high efficiency, and is therefore considered to have the potential to solve certain complex problems.

[0003] Quantum computing is currently playing an increasingly important role in solving practical problems. For example, combinatorial optimization problems involve finding an optimal solution that satisfies specific conditions from a limited set of candidate solutions. Quantum Approximate Optimization Algorithms (QAOA), a key example of quantum computing, have demonstrated remarkable potential in solving combinatorial optimization problems.

[0004] In the field of wireless communication network optimization, the Weighted Maximum Independent Set (WMIS) problem is a typical NP-Hard problem, and its solution efficiency directly affects link scheduling performance. Existing research has shown that the original QAOA, through a quantum-classical hybrid framework, has the potential to outperform classical heuristic algorithms for WMIS problems. However, the original QAOA uses a global search mechanism, and its evolution process covers both feasible and infeasible solution spaces, resulting in the following core flaws: First, quantum state evolution requires a high number of iterations (p-value) to compensate for ineffective searches, significantly increasing circuit depth and resource consumption; second, interference from infeasible solutions makes the algorithm prone to falling into local optima, especially in complex conflict graph scenarios, where convergence speed and solution quality degrade exponentially.

[0005] In response to the above challenges, the innovative breakthroughs of the present invention are reflected in the following aspects: a divide-and-conquer QUBO model is proposed, which divides the entire space into independent subspaces spanned by Dicke states of different Hamming weights, and uses XY-Mixer as a mixing operator to restrict the evolution of QAOA to these non-overlapping subspaces; a greedy algorithm and semi-positive programming are called (we extend the work of Lovász and obtain the semi-positive programming form of the maximum independent set of weighted graphs) to obtain the lower and upper bounds of the weighted maximum independent set solution, respectively; an anti-non-adiabatic driving term suitable for the evolution of the Dicke state subspace is constructed, and the core evolution formula of QAOA is modified so that it can be used as an auxiliary Hamiltonian in combination with the XY-Mixer ansatz to enable the algorithm to evolve in parallel in the independent subspaces between the upper and lower bounds, thereby accelerating convergence. Summary of the Invention

[0006] The present invention aims to solve the following technical problem: how to effectively improve the performance of quantum approximate optimization algorithm on the weighted maximum independent set problem. The function obtains the lower and upper bounds of the weighted maximum independent set, narrowing the search solution space; prepares the Dicke state as the initial state; constructs a suitable mixing operator (XY-Mixer) as the mixed Hamiltonian, so that the operator evolves in the specified subspace; then uses the nested exchange method to obtain the A(α) operator, and uses the A(α) operator to construct the anti-nonadiabatic driving term, which is added to the quantum approximate optimization algorithm circuit; uses the quantum approximate optimization algorithm to solve the weighted maximum independent set problem; this method constructs and uses the anti-nonadiabatic driving term for driving on the basis of reducing the solution space, thereby improving the solution performance of the original QAOA for solving the weighted maximum independent set problem.

[0007] The present invention solves the above technical problems through the following technical solutions, which include the following steps:

[0008] S1: Model the link scheduling in wireless communication as a conflict graph G = (V, E, W). Solving the weighted maximum independent set (WMIS) on the conflict graph G is equivalent to solving the optimal wireless link scheduling;

[0009] S2: According to the greedy algorithm, the lower bound of the weighted maximum independent set (the lower bound of the solution space) is obtained. According to the expanded The function obtains the upper bound of the largest weighted independent set (the upper bound of the solution space);

[0010] S3: Divide the entire space into independent subspaces spanned by Dicke states of different Hamming weights, use the mixing operator XY-Mixer as the mixed Hamiltonian to search in non-overlapping subspaces, and further reduce the subspaces to be searched based on upper and lower bounds;

[0011] S4: Based on Counter-Diabatic Driving, an auxiliary Hamiltonian is constructed to accelerate the evolution of the XY-Mixer.

[0012] S5: Solving the weighted maximum independent set problem based on a quantum-classical cooperative wireless link scheduling optimization method based on the divide-and-conquer QUBO model and anti-nonadiabatic drive.

[0013] Furthermore, in step S1, the conflict graph G is a weighted undirected graph. The vertices in the conflict graph G represent wireless communication links, and the edges between the vertices represent the interference generated when the wireless communication links are scheduled at the same time. The vertex weights are determined by factors such as queue length, link rate, the product of the two, or other considerations (such as fairness of the entire network). Furthermore, step S2 includes the following steps:

[0014] S21: Use a greedy algorithm to find the lower bound for WMIS. The specific steps are: select the vertex with the largest weight in the graph G; remove its edges and adjacent vertices; repeat this process on the remaining subgraph until all vertices are eliminated. The resulting set τ1 is an independent set, and |τ1| provides a lower bound on the number of vertices selected by the WMIS solution.

[0015] S22: Use semidefinite programming to solve the upper bound of WMIS. The specific steps are: Write the graph G Function form (the vertex weights of graph G are not considered at this time, that is, graph G is regarded as an unweighted graph):

[0016]

[0017] Where n is the number of vertices in the graph G, u i is the orthogonal representation of graph G, c is the unit vector in its subspace, for have (τ(G) is the maximal independent set of graph G).

[0018] Will The function is converted into a semidefinite programming form:

[0019]

[0020] Tr(B)=1

[0021]

[0022] Among them, J is an all-one matrix, B is a semidefinite matrix whose diagonal elements sum to 1; constraint B ij = 0 ensures that the matrix B complies with the adjacency structure of graph G.

[0023] Will The function is extended to the weighted graph, and the upper bound of WMIS can be obtained. For the graph G, the expanded The function is The corresponding semi-positive programming form is (reconsider the vertex weights of graph G at this time, that is, graph G is a weighted graph):

[0024]

[0025] Tr(B)=1

[0026] Among them, each element of matrix A is the maximum weight w′1 of graph G, matrix B is a semidefinite matrix, and the sum of its diagonal elements is equal to 1; constraint B ij = 0 ensures that the matrix B complies with the adjacency structure of graph G;

[0027] S23: If the upper bound of the weighted maximum independent set The difference between the lower bound (CW1) of the weighted maximum independent set is less than the minimum difference between the weights of any two vertices in the graph, that is:

[0028]

[0029] The independent set found using the greedy algorithm is the weighted maximum independent set.

[0030] Furthermore, step S3 includes the following steps:

[0031] S31: Prepare all initial states to be Dicke states, which are uniform superpositions of all binary bit strings with fixed Hamming weight k. Dicke states can be deterministically prepared using O(n) depth gates without the need for auxiliary qubits. The Dicke state has the following form:

[0032]

[0033] Where wt(x) = k means that the Hamming weight is equal to k. Where k∈{k1,…,k2}, k1=|τ1|,

[0034] S32: Constructing Ring-XY-Mixer as a Mixed Hamiltonian The Ring-XY-Mixer is constructed as follows:

[0035]

[0036] in Denote the Pauli operators X and Y, respectively, in the form of and i is the imaginary unit.

[0037] If the mixed Hamiltonian is considered as an adjacency representation of the graph, Can be regarded as Continuous quantum walk on the graph: The walker starts from a vertex on the graph and can walk within the connected subgraph where the vertex is located; if the initial state is set to a quantum superposition state with Hamming weight equal to k, then The unitary operator will confine the evolution of subsequent quantum states to this subspace;

[0038] S33: By initializing the state to the Dicke state and using the Ring-XY-Mixer, the QAOA can be made to evolve independently in the subspace defined by different k. Multiple paths from different initial states can evolve simultaneously, thereby improving efficiency. Although the traditional QUBO model is widely used in quantum solutions to combinatorial optimization problems, it has obvious limitations: it is necessary to search in the entire solution space (including feasible solutions and infeasible solutions); there is a risk of falling into local optimality, which affects the quality of the final solution. To solve these problems, the present invention proposes a divide-and-conquer QUBO model (Divided-QUBO), the core advantages of which are: decomposing the problem into non-overlapping subspaces (such as the Dicke state subspace with fixed Hamming weights), so that the algorithm evolves only in a limited solution space; further narrowing the subspace to be searched according to upper and lower bounds; through carefully designed penalty terms, solutions that do not meet the constraints can be further eliminated; the optimization of different subspaces can be performed independently, which facilitates parallel acceleration on quantum hardware.

[0039] Furthermore, step S4 includes the following steps:

[0040] S41: Constructing the Anti-Nonadiabatic Driving Term Using the Nested Exchange Method

[0041] S42: The parameterized nested exchange method is expressed as follows:

[0042] H(t)=λ(t)H T +(1-λ(t))H S

[0043]

[0044] A(α) is usually extended only to the second order, H S and H T are the initial and final Hamiltonians in the adiabatic evolution, corresponding to the mixed Hamiltonian of QAOA and the Hamiltonian of the problem The present invention uses As a mixed Hamiltonian, use As the Hamiltonian of the problem. By nested exchange, it can be concluded that A(α) is a set of operators, including many local and two-body options The design of the anti-nonadiabatic driving term must meet two objectives: first, to accelerate the evolution of the quantum state, and second, to ensure that the evolution is strictly confined to the Dicke state subspace to avoid the generation of invalid solutions. Achieving this goal faces significant theoretical challenges. Conventional methods rely on the spectral properties of the instantaneous Hamiltonian to accurately construct the driving term, but this information is often difficult to obtain in complex practical problems.

[0045] To this end, we employ a parameterized nested commutator approach, avoiding direct reliance on the instantaneous spectral properties of the Hamiltonian by approximating the anti-nonadiabatic driving term. Specifically, we systematically screen all possible operator forms, restricting the candidate to at most two-body interactions. Through theoretical derivation and numerical verification, we ultimately discover the following two operators: All of them meet the above requirements and can be used as A(α).

[0046] The inverse non-adiabatic driving term is calculated as follows:

[0047]

[0048] This is the anti-adiabatic driving term, which accelerates the convergence of the algorithm and maintains the feasible space of the solution by introducing quantum interactions that adapt to the edge structure of the graph. It is driven as an auxiliary Hamiltonian; it adjusts the local part of the current solution based on the specific edge information of the graph, and ensures that this local adjustment does not destroy the structure of the current solution.

[0049] Furthermore, step S5 includes the following steps:

[0050] S51: Set all initial states to Dicke states

[0051] S52: Constructing the Hamiltonian of the Problem Constructing Ring-XY-Mixer as a mixed Hamiltonian structure As the anti-adiabatic evolution driver; update the core evolution formula of the original QAOA to to evolve;

[0052] S53: Since the algorithm is executed in parallel, there are multiple measurement results returned, which can be compared with each other to obtain the weighted maximum independent set.

[0053] Compared with the prior art, the present invention has the following advantages:

[0054] (1) Through the greedy algorithm and the expansion function, the system can determine the lower and upper bounds of the weighted maximum independent set, thereby constraining the quantum evolution within the reduced solution space.

[0055] (2) A new divide-and-conquer QUBO model and corresponding algorithm framework are proposed. Then, the constraint-encoded mixing operator Ring-XY-Mixer is used to evolve the algorithm in a simplified subspace, which significantly improves the efficiency.

[0056] (3) Based on Counter-Diabatic Driving, an auxiliary Hamiltonian that can be used to accelerate the evolution is constructed to perform a second acceleration on the algorithm.

[0057] (4) The results of the present invention can be shown on a line graph, which shows that the present invention has a significant improvement over the original QAOA and other QAOA variant algorithms. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 This is a schematic diagram of graph structure data in Example 1 of the present invention;

[0059] Figure 2 Schematic diagram of the DQAOA core algorithm flow in the embodiment of the present invention;

[0060] Figure 3 This is a schematic diagram of the complete DQAOA process in Example 1 of the present invention;

[0061] Figure 4 This is a schematic diagram of the final feasible solution to the weighted maximum independent set problem using DQAOA in Example 1 of the present invention;

[0062] Figure 5 1 is a schematic diagram comparing the use of DQAOA to solve the weighted maximum independent set problem in Example 1 of the present invention with the original QAOA (vanilla) and other QAOA variants (Partial-mixer, DQVA, DC-QAOA, Ma-QAOA) to solve the weighted maximum independent set problem. DETAILED DESCRIPTION

[0063] The embodiments of the present invention are described in detail. This embodiment is implemented on the premise of the technical solution of the present invention, and a detailed implementation method and a specific operation process are given, but the protection scope of the present invention is not limited to the following embodiments. First, the link scheduling in wireless communication is modeled as a conflict graph G = (V, E, W). Solving the weighted maximum independent set (WMIS) on the conflict graph G is equivalent to solving the optimal wireless link scheduling. The vertices in the conflict graph G represent wireless communication links, and the edges between the vertices represent the interference generated when the wireless communication links are scheduled at the same time. The vertex weight is determined by factors such as queue length, link rate, the product of the two, or other considerations (such as the fairness of the entire network). Modeling such as Figure 1 As shown in Figure 1. The blue circles represent the vertices of the graph, the black numbers on the left inside the circles represent the vertex numbers, and the black numbers on the right represent the vertex weights. The lines between the vertices represent the existence of connecting edges between them.

[0064] For the graph G = (V, E, W), a greedy algorithm is used to obtain the lower bound of the weighted maximum independent set. The specific steps are: set the vertex set τ1 to be an empty set; sort the vertex set V in descending order of weight to obtain V′; let the vertex with the largest weight in V′ be i, add i to τ1 and delete vertex i and its adjacent nodes in V; delete the adjacent edges of i in the edge set E; repeat this process until the vertex set V is empty; the vertex set τ1 is the lower bound of the weighted maximum independent set, and the weight lower bound CW1 is the sum of the weights of all vertices in the vertex set τ1.

[0065] For graph G = (V, E, W), use semidefinite programming to solve the upper bound of WMIS. The specific steps are: write the graph G Function form:

[0066]

[0067] Where n is the number of vertices in the graph G, u i is the orthogonal representation of graph G, c is the unit vector in its subspace, for have (τ(G) is the maximal independent set of graph G).

[0068] Will The function is converted into a semidefinite programming form:

[0069]

[0070] Tr(B)=1

[0071]

[0072] Among them, J is an all-one matrix, B is a semidefinite matrix whose diagonal elements sum to 1; constraint B ij = 0 ensures that the matrix B complies with the adjacency structure of graph G.

[0073] Will By generalizing the function to the weighted graph, we can get the upper bound of WMIS. The specific steps are:

[0074] Without loss of generality, assume that {1,2,…,k} represents the maximum independent set of the unweighted graph G, so the vector {u1,u2,…,u k} are pairwise orthogonal, so we have:

[0075]

[0076] where α(G) represents the size of the largest independent set, so we have:

[0077]

[0078] Now let's reconsider the vertex weights of graph G, that is, graph G is a weighted graph. Let α′(G) represent the sum of the WMIS weights of graph G, then the following inequality holds:

[0079] w′1+w′2+…+w′ k ≥α′(G)

[0080] where w′1, w′2, …, w′ k are the weights of the first k vertices with the largest weights in graph G, and they are arranged in descending order of weight, so we have:

[0081]

[0082] After the expansion The function is defined as For graph G, the expanded The semidefinite programming form corresponding to the function is:

[0083]

[0084]

[0085] Tr(B)=1

[0086] Among them, each element of matrix A is the maximum weight w′1 of graph G, matrix B is a semidefinite matrix, and the sum of its diagonal elements is equal to 1; constraint B ij = 0 ensures that the matrix B complies with the adjacency structure of the graph G; solve the semi-positive programming form and round it down as the upper bound of the weighted maximum independent set (weight upper bound ), the corresponding number of vertices (w′1 is the maximum vertex weight in graph G).

[0087] If the difference between the upper bound of the weighted maximum independent set (θ′(G)) and the lower bound of the weighted maximum independent set (CW1) is less than the minimum difference between the weights of any two vertices in the graph, that is:

[0088]

[0089] Then the independent set found by the greedy algorithm is the weighted maximum independent set. This conclusion can be proved by contradiction: suppose τ1 is not WMIS, and another vertex set τ′1 is WMIS. If τ′1 replaces some vertices in τ1, then for i,j∈V(G), the weight difference between the added and replaced vertices must be greater than min{|w i -w j |}. This will cause the total weight of τ′1 to exceed the upper bound This leads to a contradiction. Therefore, if Then CW1 is the maximum cumulative weight and τ1 is WMIS.

[0090] All initial states are prepared as Dicke states, which are uniform superpositions of all binary bit strings with fixed Hamming weight k. Dicke states can be prepared deterministically using O(n) depth gates without the need for auxiliary qubits. The Dicke state has the following form:

[0091]

[0092] Where wt(x) = k means that the Hamming weight is equal to k. Where k∈{k1,…,k2}, k1=|τ1|, Constructing the Hamiltonian of the problem For a potential solution x, if the constraint is violated, the cumulative weight is adjusted by subtracting the penalty term:

[0093]

[0094] use Substitute and get the problem Hamiltonian The Pauli operator form of :

[0095]

[0096] After simplifying and removing the constant term, the Hamiltonian of the problem is becomes:

[0097]

[0098] At this time, the optimization goal is to maximize Convert to minimize it.

[0099] Constructing Ring-XY-Mixer as a mixed Hamiltonian The Ring-XY-Mixer is constructed as follows:

[0100]

[0101] in Denote the Pauli operators X and Y, respectively, in the form of and i is the imaginary unit.

[0102] If the mixed Hamiltonian is considered as an adjacency representation of the graph, Can be regarded as Continuous quantum walk on the graph: The walker starts from a vertex on the graph and can walk within the connected subgraph where the vertex is located; if the initial state is set to a quantum superposition state with Hamming weight equal to k, then The unitary operator will confine the subsequent evolution of the quantum state to this subspace.

[0103] By initializing the state to the Dicke state and using the Ring-XY-Mixer, the DQAOA can be made to evolve independently in the subspace defined by different k. Multiple paths from different initial states can evolve simultaneously, thereby improving efficiency. Although the traditional QUBO model is widely used in the quantum solution of combinatorial optimization problems, it has obvious limitations: when encoding constraints, the QUBO model usually relies on penalty terms to transform the constrained optimization problem into an unconstrained form. However, this processing method requires the algorithm to search in the entire solution space (including feasible solutions and infeasible solutions), resulting in a waste of computing resources and a slow convergence rate; the introduction of penalty terms may complicate the energy level structure of the Hamiltonian, increase the risk of the algorithm falling into a local optimum, and affect the quality of the final solution. To address these issues, the present invention proposes the Divided-QUBO model, whose core advantages are: by decomposing the problem into non-overlapping subspaces (such as the Dicke state subspace with fixed Hamming weights), the algorithm evolves only within a limited solution space, greatly reducing invalid searches; although infeasible solutions may still exist in the subspace, through carefully designed penalty terms, solutions that do not meet the constraints can be further eliminated, thereby improving the solution efficiency; the optimization of different subspaces can be performed independently, facilitating parallel acceleration on quantum hardware.

[0104] The design of the anti-nonadiabatic driving term must meet two objectives: first, to accelerate the evolution of the quantum state, and second, to ensure that the evolution is strictly confined to the Dicke state subspace to avoid the generation of invalid solutions. Achieving this goal faces significant theoretical challenges. Conventional methods rely on the spectral properties of the instantaneous Hamiltonian to accurately construct the driving term, but this information is often difficult to obtain in complex practical problems.

[0105] To this end, we employ a parameterized nested commutator approach, avoiding direct reliance on the instantaneous spectral properties of the Hamiltonian by approximating the anti-nonadiabatic driving term. Specifically, we systematically screen all possible operator forms, restricting the candidate to at most two-body interactions. Through theoretical derivation and numerical verification, we ultimately discover the following two operators: All of them meet the above requirements and can be used as A(α).

[0106] The parameterized nested exchange method is expressed as follows:

[0107] H(t)=λ(t)H T +(1-λ(t))H S

[0108]

[0109] HS and H T are the initial and final Hamiltonians in the adiabatic evolution, corresponding to the mixed Hamiltonian of DQAOA and the Hamiltonian of the problem The present invention uses As a mixed Hamiltonian, use as the problem Hamiltonian. In general, A(α) is truncated at the second order to avoid nonlocal multi-body interactions, limiting the candidate anti-nonadiabatic driving terms to at most two-body interactions. This approach strikes a balance between implementation complexity and approximation accuracy. By nesting the exchange, it can be shown that A(α) is a set of operators containing many local and two-body candidates. The selection of the anti-adiabatic driving term needs to balance the acceleration of the algorithm evolution and the limitation of the evolution to the subspace of the Dicke state, which requires some heuristic techniques. After derivation, Both operators meet the above requirements and can be used as A(α).

[0110] The inverse non-adiabatic driving term is calculated as follows:

[0111]

[0112] This is the anti-adiabatic driving term, which accelerates the convergence of the algorithm and maintains the feasible space of the solution by introducing quantum interactions that adapt to the edge structure of the graph. It is driven as an auxiliary Hamiltonian; it adjusts the local part of the current solution based on the specific edge information of the graph, and ensures that this local adjustment does not destroy the structure of the current solution.

[0113] Initialize SW′=SW=0 to represent two consecutive measurement results; update the original QAOA core evolution formula to Evolve and use COBYLA optimizer to optimize α, β, γ. Measure the quantum state |α, β, γ> and we can get The expected value SW on |α,β,γ>, where If the absolute value of the difference between SW and SW′ is less than a threshold ∈, the measurement result is returned. Otherwise, SW is assigned to SW′, and the COBYLA optimizer is used to optimize α, β, and γ until the measurement result is returned. The algorithm executes in parallel and returns multiple measurement results, which are compared to obtain the weighted maximum independent set.

[0114] Using the DQAOA algorithm, Evolve, use COBYLA optimizer to optimize α, β, γ, and measure the quantum state |α, β, γ> to obtain the expected value SW, where If the absolute value of the difference between SW and SW′ is less than a threshold ∈, the measurement result is returned. Otherwise, SW is assigned to SW′, and the COBYLA optimizer is used to optimize α, β, and γ until the measurement result is returned. The algorithm executes in parallel and returns multiple measurement results, which are compared to obtain the weighted maximum independent set.

[0115] The core algorithm structure of DQAOA is as follows: Figure 2 As shown, the overall structure is as Figure 3 In this example, the feasible solution is as follows Figure 4 shown.

[0116] The original QAOA is mainly used to solve classic combinatorial optimization problems. Compared with the original QAOA method, the DQAOA method proposed in this invention has significantly improved performance. The comparison results of DQAOA with the original QAOA (vanilla) and other QAOA variants are shown in the figure. Figure 5 As shown in the figure. QAOA variant algorithms include Partial-mixer, DQVA, DC-QAOA, and Ma-QAOA. They are introduced as follows:

[0117] (1) Partial-mixer

[0118] The partial mixing scheme is given by Hadfield and is defined as:

[0119]

[0120] where j∈V(G), D j is the degree of vertex j, i∈nbhd(j) refers to the adjacent nodes of j. The solution is set as:

[0121] |s>=|0…01>

[0122]

[0123] (2) DQVA (Dynamic Quantum Variational Algorithm)

[0124] Dynamical Quantum Variational Ansatz (DQVA) is based on partial mixing

[0125] The scheme gives each partial mixing operator H CX,j A separate parameter is assigned, which is defined as follows:

[0126]

[0127] The initial state is prepared as W state:

[0128]

[0129] Build as:

[0130]

[0131] Algorithm evolution can be described as:

[0132]

[0133] In this scheme, the initial state is not a uniform superposition of all feasible solutions, but a uniform superposition of all quantum states with Hamming weight 1. In order to fully mix, the algorithm first performs

[0134] (3) DC-QAOA (Digital Inverse Non-adiabatic Evolution QAOA)

[0135] The Digital Counter-Diabatic QAOA (DC-QAOA) is based on the original QAOA and provides an improved QAOA algorithm with a CD term. The CD term is By embedding the auxiliary Hamiltonian in the original QAOA Promoting the rapid evolution of the algorithm towards the final state:

[0136]

[0137] Among them C n is a complete graph with n vertices. The final state can be evolved according to the following formula:

[0138]

[0139] (4)Ma-QAOA (multi-angle QAOA)

[0140] Multi-angle QAOA in mixed Hamiltonian and the Hamiltonian of the problem Introducing additional parameters:

[0141]

[0142] In summary, the present invention discloses a quantum-classical cooperative wireless link scheduling optimization method based on the divide-and-conquer QUBO model and anti-nonadiabatic drive. Based on the original QAOA method, the present invention proposes a new divide-and-conquer QUBO model and the corresponding algorithm framework. The lower and upper bounds of the optimal solution are determined by using the greedy algorithm and semidefinite programming. Then, the constraint-coded hybrid operator Ring-XY-Mixer is used to evolve the algorithm in a simplified subspace, which significantly improves the efficiency. The semidefinite programming method of functions with respect to maximum independent sets (MIS) is extended to weighted graphs, making it possible to determine the upper bound of weighted maximum independent sets; inspired by anti-adiabatic driving, an auxiliary Hamiltonian is constructed to accelerate the evolution of Ring-XY-Mixer.

[0143] Although the embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention.

Claims

1. A quantum-classical collaborative wireless link scheduling optimization method based on the divide-and-conquer QUBO model and anti-nonadiabatic drive, characterized by: The following steps are involved: S1: Model the link scheduling in wireless communication as a conflict graph G = (V, E, W). Solving the weighted maximum independent set (WMIS) on the conflict graph G is equivalent to solving the optimal wireless link scheduling; S2: Obtain the lower bound of the weighted maximum independent set (the lower bound of the solution space) according to the greedy algorithm, and obtain the upper bound of the weighted maximum independent set (the upper bound of the solution space) according to the expanded Lovász-θ function; S3: Divide the entire space into independent subspaces spanned by Dicke states of different Hamming weights, use the mixing operator XY-Mixer as the mixed Hamiltonian to search in non-overlapping subspaces, and further reduce the subspaces to be searched based on upper and lower bounds; S4: Based on Counter-Diabatic Driving, an auxiliary Hamiltonian is constructed to accelerate the evolution of the XY-Mixer. S5: Run the quantum-classical cooperative wireless link scheduling optimization method based on the divide-and-conquer QUBO model and anti-nonadiabatic drive to solve the weighted maximum independent set problem.

2. A quantum-classical collaborative wireless link scheduling optimization method based on the divide-and-conquer QUBO model and anti-nonadiabatic drive according to claim 1, characterized in that: In step S1, the conflict graph G is a weighted undirected graph, the vertices in the conflict graph G represent wireless communication links, and the edges between the vertices represent the interference generated when the wireless communication links are scheduled at the same time; the vertex weights are determined by factors such as queue length, link rate, the product of the two, or other considerations (such as the fairness of the entire network).

3. A quantum-classical collaborative wireless link scheduling optimization method based on the divide-and-conquer QUBO model and anti-nonadiabatic drive according to claim 1, characterized in that: The step S2 comprises the following steps: S21: Use a greedy algorithm to find the lower bound of WMIS. The specific steps are: select the vertex with the largest weight in the graph G; remove its edges and adjacent vertices; repeat this process on the remaining subgraph until all vertices are eliminated; the resulting set τ1 is an independent set, and |τ1| provides a lower bound on the number of vertices selected by the WMIS solution; S22: Use semidefinite programming to solve the upper bound of WMIS. The specific steps are: Write the Lovász-θ function form of graph G (the vertex weights of graph G are not considered at this time, that is, graph G is regarded as an unweighted graph): Where n is the number of vertices in the graph G, u i is an orthogonal representation of the graph G, c is a unit vector in its subspace, and for θ(G), θ(G)>|τ(G)| (τ(G) is the maximal independent set of the graph G); Convert the Lovász-θ function into a semidefinite programming form: Tr(B)=1 B≥0 Among them, J is an all-one matrix, B is a semidefinite matrix whose diagonal elements sum to 1; constraint B ij = 0 ensures that the matrix B complies with the adjacency structure of graph G; By generalizing the Lovász-θ function to weighted graphs, we can obtain the upper bound of WMIS. For graph G, the semi-positive programming form corresponding to the extended Lovász-θ function is (now reconsidering the vertex weights of graph G, that is, graph G is a weighted graph): B≥0 Tr(B)=1 Among them, each element of the matrix A is the maximum weight w1 of the graph G ′ , the matrix B is a semidefinite matrix, the sum of its diagonal elements is equal to 1; constraint B ij = 0 ensures that the matrix B complies with the adjacency structure of graph G; S23: If the difference between the upper bound θ′(G) of the weighted maximum independent set and the lower bound CW1 (the sum of the weights of all vertices in τ1) of the weighted maximum independent set is less than the minimum difference between the weights of any two vertices in the graph, that is: θ ′ (G)-CW1 <min{|w i -w j |} The independent set found using the greedy algorithm is the weighted maximum independent set.

4. A quantum-classical collaborative wireless link scheduling optimization method based on the divide-and-conquer QUBO model and anti-nonadiabatic drive according to claim 1, characterized in that: The step S3 includes the following steps: S31: Prepare all initial states to be Dicke states, which are uniform superpositions of all binary bit strings with fixed Hamming weight k. Dicke states can be deterministically prepared using O(n) depth gates without auxiliary qubits. The form of the Dicke state is as follows: Where wt(x) = k means that the Hamming weight is equal to k, where k∈{k1,…,k2}, k1 = |τ1|, S32: Constructing Ring-XY-Mixer as a mixed Hamiltonian The Ring-XY-Mixer is constructed as follows: in Denote the Pauli operators X and Y, respectively, in the form of and i is the imaginary unit; If the mixed Hamiltonian is considered as an adjacency representation of the graph, Can be regarded as Continuous quantum walk on the graph: The walker starts from a vertex on the graph and can walk within the connected subgraph where the vertex is located; if the initial state is set to a quantum superposition state with Hamming weight equal to k, then The unitary operator will confine the evolution of subsequent quantum states to this subspace; S33: By initializing the state to the Dicke state and using the Ring-XY-Mixer, the QAOA can be made to evolve independently in the subspace defined by different k; multiple paths from different initial states can evolve simultaneously, thereby improving efficiency; although the traditional QUBO model is widely used in the quantum solution of combinatorial optimization problems, it has obvious limitations: it is necessary to search in the entire solution space (including feasible solutions and infeasible solutions); there is a risk of falling into local optimality, which affects the quality of the final solution; to solve these problems, the present invention proposes a divide-and-conquer QUBO model (Divided-QUBO), whose core advantages are: decomposing the problem into non-overlapping subspaces (such as the Dicke state subspace with fixed Hamming weights), so that the algorithm evolves only in a limited solution space; further narrowing the subspace to be searched according to the upper and lower bounds; through carefully designed penalty terms, solutions that do not meet the constraints can be further eliminated; the optimization of different subspaces can be performed independently, which facilitates parallel acceleration on quantum hardware.

5. A quantum-classical collaborative wireless link scheduling optimization method based on the divide-and-conquer QUBO model and anti-nonadiabatic drive according to claim 1, characterized in that: The step S4 comprises the following steps: S41: Constructing the Anti-Nonadiabatic Driving Term Using the Nested Exchange Method S42: The nested exchange method is expressed as follows: H(t)=λ(t)H T +(1-λ(t))H S A(α) is usually extended only to the second order, H S and H T are the initial and final Hamiltonians in the adiabatic evolution, corresponding to the mixed Hamiltonian of QAOA and the Hamiltonian of the problem The present invention makes As a mixed Hamiltonian, use As the Hamiltonian of the problem; through nested exchange, it can be concluded that A(α) is a set of operators, including many local and two-body options The design of the anti-nonadiabatic driving term must meet two objectives: first, to accelerate the evolution of the quantum state, and second, to ensure that the evolution is strictly confined to the Dicke state subspace to avoid the generation of invalid solutions. Achieving this goal faces significant theoretical challenges. Conventional methods rely on the spectral properties of the instantaneous Hamiltonian to accurately construct the driving term, but such information is often difficult to obtain in complex practical problems. To this end, we employ a parameterized nested commutator approach to avoid relying directly on the instantaneous spectral properties of the Hamiltonian by approximating the anti-nonadiabatic driving term. Specifically, we systematically screen all possible operator forms, restricting the candidate to at most two-body interactions. Through theoretical derivation and numerical verification, we ultimately discover the following two operators: All of them meet the above requirements and can be used as A(α); The calculation formula of the anti-adiabatic driving term is: α ij ∈[0,π]; That is, the anti-adiabatic driving term, which accelerates the convergence of the algorithm and maintains the feasible space of the solution by introducing quantum interactions adapted to the graph edge structure; the present invention selects Driven by an auxiliary Hamiltonian, it makes local adjustments to the local part of the current solution based on the specific edge information of the graph, and ensures that this local adjustment does not destroy the structure of the current solution.

6. A quantum-classical collaborative wireless link scheduling optimization method based on the divide-and-conquer QUBO model and anti-nonadiabatic drive according to claim 1, characterized in that: The step S5 comprises the following steps: S51: Set all initial states to Dicke states S52: Constructing the Hamiltonian of the Problem Constructing Ring-XY-Mixer as a mixed Hamiltonian structure As the anti-adiabatic evolution driver; the core evolution formula in the original QAOA is updated to to evolve; S53: Since the algorithm is executed in parallel, there are multiple measurement results returned, which can be compared with each other to obtain the weighted maximum independent set.

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