A single-path traffic transmission optimization method and system for complex network

Abstract

The application discloses a kind of single-path traffic transmission optimization methods and systems for complex network, it is related to network resource scheduling and quantum computing application technical field, including the following steps: the maximum transmission flow of each feasible path is dispersed into multiple flow levels, and the bias coefficient is dispersed into multiple bias coefficient levels;Log-level quantum bits are used to binary encode feasible path, flow level and bias coefficient level respectively, to obtain corresponding multiple encoding variables;Multiple encoding variables are fitted, and a QUBO model with cost minimization and flow maximization as optimization objectives is constructed;Different multiple encoding variables and auxiliary binary variables are input into the QUBO model, and the QUBO model is solved based on quantum optimization algorithm to obtain the optimal result.The application meets the requirement that the main flow sub-optimization algorithm is the highest order of decision variable is quadratic, significantly reduces the calculation redundancy caused by repeated optimization of multiple parameters in traditional method.

Description

Technical Field

[0001] This invention relates to the fields of network resource scheduling and quantum computing application technology, and in particular to a method and system for optimizing single-path traffic transmission in complex networks. Background Technology

[0002] Single-path traffic transmission optimization is one of the core problems in complex network traffic scheduling and routing optimization, and it is widely used in key engineering applications such as power system flow analysis, data center network optimization, communication network parameter configuration, traffic route planning, and supply chain logistics routing. The core of this problem is to select a feasible path between the source node and the sink node, optimize the transmission traffic under the constraint of path capacity, and achieve a comprehensive trade-off between minimizing costs and maximizing traffic.

[0003] As network scale expands and topological complexity increases, single-path transmission optimization faces two core challenges: First, the number of feasible paths grows exponentially, and the decision variables of traditional optimization algorithms increase linearly with network scale, easily leading to combinatorial explosion in high-dimensional decision spaces, making it difficult to obtain the optimal solution in polynomial time. Second, there is a strong nonlinear coupling relationship between the unit traffic cost of a path and the transmission traffic. This nonlinearity stems from factors such as resource congestion, capacity constraints, and transmission losses, making it difficult to accurately model using a unified function, resulting in the difficulty of achieving synergistic optimization between cost and traffic.

[0004] Quantum computing, with its advantages of superposition and parallelism, offers new solutions to high-dimensional combinatorial optimization problems. Quantum approximate optimization algorithms (QAOA) and quantum annealing, among others, can achieve exponential speedups on specific problems. However, the number of qubits in existing quantum algorithms is linearly tied to network size and the number of paths. Furthermore, excessive optimization parameters in the model fail to meet the requirement of mainstream quantum optimization algorithms that the highest order of decision variables is quadratic. This leads to redundant optimization of multiple parameters, consuming significant computational resources and resulting in computational redundancy. Summary of the Invention

[0005] This invention provides a method and system for optimizing single-path traffic transmission in complex networks, which solves the existing problems.

[0006] In a first aspect, the present invention provides a method for optimizing single-path traffic transmission in complex networks, comprising the following steps:

[0007] Obtain all feasible paths from the source node to the sink node in a directed connected network, the maximum transmission flow of each path, and the bias coefficient used to balance the nonlinear relationship between cost and flow.

[0008] The maximum transmission traffic of each feasible path is discretized into multiple traffic levels, and the bias coefficient is discretized into multiple bias coefficient levels. Logarithmic-level qubits are used to binary encode the feasible path, traffic level, and bias coefficient level respectively, resulting in multiple corresponding encoded variables. The multiple encoded variables are fitted to construct a QUBO model with the optimization objectives of cost minimization and traffic maximization. In this model, the coupling terms of multiple encoded variables are equivalently transformed into quadratic terms through auxiliary binary variables.

[0009] By inputting multiple different encoded variables and auxiliary binary variables into the QUBO model, and solving the QUBO model based on the quantum optimization algorithm, the optimal transmission path, optimal transmission flow, and corresponding cost are obtained.

[0010] Preferably, the step of using logarithmic-level qubits to perform binary encoding on feasible paths, flow rates, and bias coefficients to obtain multiple corresponding encoded variables specifically includes:

[0011] use A feasible path for each quantum bit pair p Encode the path to obtain the path-encoded variable. r k ;

[0012] use Each qubit encodes the flow rate level, resulting in a flow rate encoded variable. d i , L The number of traffic levels;

[0013] use Each qubit encodes a level of bias coefficient, resulting in a bias coefficient encoded variable. a j , M This represents the number of bias coefficient levels.

[0014] Preferably, the step of fitting multiple encoded variables to construct a QUBO model with the optimization objectives of minimizing cost and maximizing traffic specifically includes the following steps:

[0015] The path-coded variables and flow-coded variables are combined into a binary variable vector. Based on the binary variable vector, the cost function and flow function are mapped to a combination of linear and quadratic terms of the binary variables through least squares fitting, thus obtaining the corresponding fitting expression.

[0016] The loss function is constructed based on the parameter vector in the fitting expression. The gradient of the loss function with respect to the parameter vector is set to 0. The specific values ​​of the parameter vector are obtained and substituted into the fitting expression to obtain the corresponding QUBO expression.

[0017] The bias coefficient encoding variables are weighted and summed based on the value range of the bias coefficient to obtain discrete values ​​of the bias coefficient. The discrete values ​​of the bias coefficient are then combined with the QUBO expression to obtain the QUBO model.

[0018] Preferably, the step of converting the coupling terms of multiple encoded variables in the QUBO model into equivalent quadratic terms using auxiliary binary variables specifically includes the following steps:

[0019] Introduce an auxiliary binary variable and add a penalty term with a penalty coefficient to the QUBO model, which forces the auxiliary binary variable to be equal to the product of the two sets of variables in the coupling term;

[0020] The coupling terms are transformed into standard QUBO form based on auxiliary binary variables and penalty terms.

[0021] Preferably, the QUBO model is as follows:

[0022] ;

[0023] In the formula, r k For the first k Path-encoded variables, d i For the first i Each flow-encoded variable a j For the first j Each bias coefficient encodes a variable. y For auxiliary binary variables, including the first auxiliary variable y jk With the second auxiliary variable y ji , These are the discrete values ​​of the bias coefficient. This is the quadratic form of the QUBO expression for the cost function. This is the quadratic form of the QUBO expression for the flow function. and There are two penalty items.

[0024] Preferably, the step of solving the QUBO model based on the quantum optimization algorithm to obtain the optimal transmission path, optimal transmission flow, and corresponding cost specifically includes the following steps:

[0025] The QAOA algorithm is used to solve the QUBO model to obtain the selection result, namely the optimal discrete combination of path-flow-bias coefficient;

[0026] The selection results are refined by spline interpolation to obtain the optimal transmission path, optimal transmission flow, and corresponding cost.

[0027] Preferably, the step of refining the screening results through spline interpolation to obtain the optimal transmission path, optimal transmission traffic, and corresponding cost specifically includes the following steps:

[0028] Determine a local optimization interval centered on the screening results, and select discrete sample points within the interval;

[0029] Construct a cubic spline interpolation function that satisfies the continuity of function values, first derivative, and second derivative at discrete sample points;

[0030] By solving for the minimum value of the interpolation function through extreme value analysis, the optimal transmission path, optimal transmission flow, and corresponding cost can be obtained.

[0031] Preferably, the cubic spline interpolation function is as follows:

[0032] ;

[0033] In the formula, It is a cubic spline interpolation function. , , , These are the interpolation coefficients. For discrete sample points, To and The flow rate value of the next adjacent discrete sample point. This represents the flow rate.

[0034] Preferably, a depth-first search algorithm is used to obtain all feasible paths from the source node to the sink node in the directed connected network.

[0035] Secondly, the present invention also provides a single-path traffic transmission optimization system for complex networks, comprising:

[0036] The acquisition module is used to acquire all feasible paths from the source node to the sink node in a directed connected network, the maximum transmission flow of each path, and the bias coefficient used to balance the nonlinear relationship between cost and flow.

[0037] A module is constructed to discretize the maximum transmission traffic of each feasible path into multiple traffic levels and the bias coefficient into multiple bias coefficient levels. Logarithmic-level qubits are used to binary encode the feasible path, traffic level, and bias coefficient level respectively, resulting in multiple corresponding encoded variables. The multiple encoded variables are fitted to construct a QUBO model with the optimization objectives of minimizing cost and maximizing traffic. In this model, auxiliary binary variables are used to equivalently transform the coupling terms of multiple encoded variables into quadratic terms.

[0038] The solution module is used to input multiple different encoded variables and auxiliary binary variables into the QUBO model, and solve the QUBO model based on the quantum optimization algorithm to obtain the optimal transmission path, optimal transmission flow and corresponding cost.

[0039] Compared with the prior art, the beneficial effects of the present invention are:

[0040] This invention employs logarithmic-level qubit encoding to encode path, flow, and bias coefficients, transforming the relationship between the number of qubits and network size from linear to logarithmic. This significantly reduces the quantum computing resource requirements for large-scale complex networks, making quantum optimization algorithms applicable to such networks. Furthermore, by embedding path, flow, and bias coefficients into the same QUBO model and utilizing auxiliary binary variables to equivalently transform higher-order terms resulting from multi-parameter coupling into quadratic terms, this satisfies the requirement of mainstream quantum optimization algorithms that the highest order of decision variables is quadratic. This achieves one-time collaborative optimization of multiple optimization parameters, significantly reducing computational redundancy caused by repeated multi-parameter optimization in traditional methods. Attached Figure Description

[0041] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0042] Figure 1 This is a flowchart of a single-path traffic transmission optimization method for complex networks according to the present invention.

[0043] Figure 2 This is a schematic diagram of directed network traffic transmission according to the present invention;

[0044] Figure 3 This is a schematic diagram of the QAOA quantum circuit for the quantum optimization model of this invention;

[0045] Figure 4 The structure of the Internet of Things (IoT) communication network according to an embodiment of the present invention;

[0046] Figure 5 The calculation results of the QAOA algorithm in this embodiment of the invention;

[0047] Figure 6 This is the QAOA optimization result of an embodiment of the present invention. Detailed Implementation

[0048] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0049] A single-path traffic transmission optimization method for complex networks is presented, specifically a quantum-classical cooperative single-path transmission optimization method. Its overall process includes complex network modeling, QUBO (Quadratic Unconstrained Binary Optimization) model construction, quantum coarse screening, and classical fine-tuning optimization. (Refer to...) Figure 1 Specifically, it includes the following steps:

[0050] S1: Construct a single-path transmission optimization model for complex networks.

[0051] For directed connected networks G ( V , E ),in, V For a set of nodes, E For a directed edge set, each edge ( m , n )∈ E With maximum capacity u mn Cost per unit of traffic c mn Specify the source node s and the sink node t, such as... Figure 2 As shown. Using existing classical search methods, such as the Depth-First Search (DFS) algorithm, all feasible paths from s to t are extracted. Let the number of feasible paths be . p The maximum capacity of each path is determined by the edge with the smallest capacity among all edges contained in the path; this is known as the "barrel effect." (See reference...) Figure 2 The maximum capacity of each path is f p,max Each path p The unit traffic cost is c p .

[0052] The cost per unit of traffic exhibits a non-linear relationship with traffic volume: under high load, costs rise rapidly due to increased losses, while the relationship deviates from linearity even under low load. This non-linearity is difficult to model accurately at the coarse-grained level, but can be calculated using specific traffic values ​​and addressed in fine-grained optimization. Generally, the traffic volume of a path is positively correlated with the cost per unit of traffic. Prioritizing low-cost paths limits the maximum transmission traffic, while pursuing high traffic volume leads to an increase in total cost. Since the non-linear relationship between path traffic and cost is difficult to describe with a unified function, a systematic optimization framework is needed to achieve a trade-off between traffic and cost. This is addressed by introducing a cost-traffic bias parameter. l The dual-objective problem of minimizing cost and maximizing flow is transformed into a single-objective optimization problem. Cost and flow are normalized to eliminate dimensional differences and improve numerical stability. The optimization objective and constraints are as follows:

[0053] (1);

[0054] in, F ( p ) is the path p The flow function of transmission C ( p ) is the path p The cost function, E P For path p Contained edges, f p For path p Transmission traffic, For path p Maximum transmission traffic l The value range of is [0,1]. l As the value approaches 1, the optimization objective tends to minimize the pure cost. l When the value approaches 0, the optimization objective tends towards maximizing pure flow, but if l If the value is too high or too low, the model will degenerate into an extreme case, causing the path selection problem to lose its multi-objective trade-off significance. Therefore, this invention needs to set... l The feasible range of values ​​is [ l min , l max The value of [[] is determined by the application scenario, the dimensions and magnitudes of cost and traffic. The significant differences in dimensions and magnitudes between the original cost and traffic can easily widen the coefficient range and reduce numerical stability, necessitating normalization. Let the original cost and traffic ranges be [[]]. c min , c max ]and[ f min , fmax Its normalized form is defined as:

[0055] (2);

[0056] in, c The cost value before normalization. f The flow rate before normalization. c ∗ , f ∗ For the normalized dimensionless cost and flow variables, c ∗ , f ∗ ∈[0,1].

[0057] S2: Quantum low-dimensional encoding and QUBO model construction.

[0058] To address the issue of linear growth in the number of qubits, logarithmic qubit encoding is employed to binary encode feasible paths, flow levels, and cost-flow bias coefficients, achieving exponential compression of the decision space. Firstly, a... One quantum bit, through binary variables r k Combinatorial representation of ∈0,1 p Index the feasible paths; then calculate the maximum traffic for each path. f p,max Discretized L Each level is adopted. One quantum bit, through binary variables d i ∈0,1 represents the traffic level; finally, the bias coefficient is encoded, which... l interval [ l min , l max Discretized into M Each level, through binary variables a j ∈0,1 indicates the bias coefficient level.

[0059] Path-encoded variables r k Flow coding variables d i Merged into a combined binary variable vector x:

[0060] x=[ r 1,…, r k , d 1,…, d i (3);

[0061] The cost and flow functions are mapped to linear and quadratic combinations of binary variables using least squares fitting. Based on the normalized cost and flow in formula (2), the coefficients of QUBO are fitted. The fitting formula is:

[0062] (4);

[0063] in, Let be the fitting cost function for the two variables. The fitted flow function for the two variables is a vector consisting of the normalized cost and flow in formula (2). and Let be symmetric matrices, representing the quadratic coefficient vectors of the cost and flow fitting terms, respectively. C With q F The vector of coefficients for linear terms. and A vector of constant terms. C As a cost variable, F For flow rate variables.

[0064] To represent the cost function corresponding to the path flow combination in a quadratic form using binary variables, this invention employs least squares regression to fit it. Let θ be the parameter vector to be determined, and the fitting error is determined by the loss function. L (θ) is used to represent this, as shown below:

[0065] (5);

[0066] Where A is the observation matrix, which consists of all variable terms corresponding to the path flow combination; This is the cost vector corresponding to each combination. To obtain the optimal parameter θ... * This invention calculates the gradient of the loss function with respect to θ. And set it to zero:

[0067] (6);

[0068] In the formula, T This is the transpose operator.

[0069] The analytical solution for the parameter vector θ is:

[0070] (7);

[0071] Find θ * Then, the coefficients of the linear and quadratic terms can be obtained:

[0072] (8);

[0073] in, h and J The fitted value corresponds to matrix q in equation (4). C and The value, k and i These represent indices indicating different paths and traffic levels. α and β The subscript is used to distinguish different binary variables in the quadratic coefficient fitting. Subscripts containing "c" indicate that these are coefficients of the cost function. The path to be found k The coefficients of the first-order term in the fitted cost term, For the path in the parameter vector k The first-order fitting coefficient of the cost item, For the desired flow level i The coefficients of the first-order term in the fitted cost term, For the flow level in the parameter vector i The first-order fitting coefficient of the cost item, The path to be found k Two different binary variables α and β The quadratic coefficient of the fitting cost term, For the path in the parameter vector k Two different binary variables in α and β The quadratic coefficient of the fitting cost term, For the desired flow level i Two different binary variables α and β The quadratic coefficient of the fitting cost term, For the flow level in the parameter vector i Two different binary variables in α and β The quadratic coefficient of the fitting cost term, The path to be found k With traffic level i The fitting cost coefficient of the binary variables in the data. For the path in the parameter vector k With traffic level i The fitting cost coefficient of the binary variables in the data. The value of the last column of θ.

[0074] It is worth noting that when using k When a path is represented by a combination of binary variables, if 2 k > pThe excess binary combinations correspond to physically infeasible paths. To prevent the optimizer from selecting these infeasible paths during evolution, a penalty term needs to be introduced. .

[0075] (9);

[0076] in, S N Indicates an invalid path index set. Index for an invalid path. m N,c The value of is the penalty coefficient, which must be significantly larger than the fitted value of the QUBO coefficient mentioned above, i.e., the values ​​of Q, q and c in equation (4). This term can be regarded as adding an extra linear penalty coefficient to ensure that the quantum amplitude of infeasible paths is significantly suppressed during the optimization process, so that quantum optimization is only completed within the set of physically reachable paths.

[0077] After fitting, the QUBO (Quadratic Unconstrained Binary Optimization) form of the cost function in equation (4) is written as follows: C ( r , d Its expression is:

[0078] (10);

[0079] In the formula, and To represent a path k binary variable, subscript α and β Used to distinguish paths k Different binary variables, and To represent the flow level i binary variable, subscript α and β Used to distinguish and represent traffic levels i Different binary variables.

[0080] Similarly, the function representing flow rate F ( r , d The QUBO expression for ) can also be obtained through the same fitting process:

[0081] (11);

[0082] In the formula, The path to be found k The coefficients of the first-order term in the fitted flow term, For the desired flow level i The coefficients of the first-order term in the fitted flow term, The path to be found k Two different binary variables α and β The quadratic coefficients of the fitted flow term, For the desired flow level i Two different binary variables α and β The quadratic coefficients of the fitted flow term, The path to be found k With traffic level i Fitted flow coefficients of binary variables in the data.

[0083] Among these coefficients, the subscripts include f This indicates that the coefficients of the flow function are...

[0084] Through the fitting process described above, the effects of path selection and flow rate have been uniformly represented as a combination of linear and quadratic terms of binary variables. However, in multi-objective optimization, cost and flow rate terms still need to be balanced using bias coefficients.

[0085] In this invention, the following is introduced from equation (1) l They are incorporated into the QUBO framework as optimization variables and modeled in discrete form. This discretization process maintains consistency with the previous binary representation of path and flow variables. They are uniformly distributed... l Divided into M There are several discrete levels. To efficiently encode these discrete levels, a set of binary variables is introduced. a ={ a 1, a 2,…, a Y},in . l The discrete values ​​are represented using a binary weighted sum, expressed as:

[0086] (12);

[0087] Based on the above statements l It is no longer a parameter that can be continuously adjusted, but a discrete tradeoff coefficient uniquely determined by a set of binary variables. j Subscripts for different bias levels.

[0088] Next, the present invention will l Combination with path traffic ( r , d Combining these elements, the weight of the flow fitting term in the objective function is changed from... l(a) Regulation. With l The objective function of the combined path flow optimization model is:

[0089] (13);

[0090] This optimization model can naturally satisfy the constraints, but the objective function inevitably contains forms of... a j ·r k ·d i The third-order term (coupling term) forms a Higher Order Binary Optimization (HOBO) form. Currently, mainstream quantum optimization algorithms can only support quadratic terms at most, so this model cannot be solved directly. Therefore, it is necessary to equivalently convert the third-order term into a quadratic form. This is achieved by introducing an auxiliary binary variable. y Furthermore, a penalty term with a large penalty coefficient G is added to the objective function, which forces... y equal a j ·r k ·d i The product of the two sets of variables. Through this construction, the optimal solution to the original problem remains unchanged. Conversely, combinations of variables that do not conform to this relationship will incur additional energy penalties and therefore will not appear in the optimal solution. Thus, the original HOBO model with third-order terms can be systematically converted into the standard QUBO form. Specifically, this invention introduces auxiliary binary variables. y jk and y ji Then the third item a j ·r k ·d i The minimization problem can be equivalently expressed as:

[0091] (14);

[0092] in, and When converting a cubic term to a quadratic term, the penalty term for the auxiliary binary variable not being equal to the two set binary variables is applied.

[0093] Finally, the complete QUBO model is shown below:

[0094] (15);

[0095] in, C * ( r , d , y )and F * ( r,d,y ) respectively represent in all a j ·r k ·d i Item replaced with y jk ·d i or y ji ·r k back C ( r , d )and F ( r , d The quadratic form of ) is given by the decision variable [ r 1,…, r K , d 1,…, d I , a 1,…, a Y , y 11 ,…, y jk , y ji ].

[0096] After Equation (1) is transformed into Equation (15), it conforms to the standard QUBO form and can be directly solved using a quantum optimization algorithm. The decision variables are [ r 1,…, r K , d 1,…, d I , a 1,…, a J , y 11 ,…, y jk , y ji ].

[0097] After simplification, the number of variables in the QUBO model is: O ( j (k + i Compared with traditional algorithms O (2 j+k And other quantum algorithms (where the number of qubits is linearly related to the number of paths) O ( p In contrast, the method of this invention shows that the decision space shrinks exponentially. This optimization model can explore various operational preferences. The final solution is determined by selecting a combination of objective functions with minimum values ​​under different bias coefficients, resulting in a path flow scheme.

[0098] S3: Global coarse filtering based on QAOA.

[0099] The aforementioned optimization model corresponds one-to-one with the Ising Hamiltonian and can be directly embedded into the QAOA framework for solution. QAOA searches for the optimal solution in the superposition state space through a parameterized sequence of quantum gates, naturally biasing the probability distribution towards lower energy states, thus obtaining paths with lower cost and higher throughput under different bias coefficients. QAOA achieves quantum interference-driven global search by alternately applying cost Hamiltonians and hybrid Hamiltonians in the quantum state space. Let the QUBO model contain... N There are two binary decision variables, corresponding to N Each qubit represents a decision variable. x i ∈{0,1}. During algorithm initialization, a Hadamard gate is applied to all qubits to generate a uniform superposition state:

[0100] (16);

[0101] in, Represents the initial quantum uniform superposition state. It is the direct product symbol. N The direct product of a superposition of single-bit states.

[0102] This state covers all possible path-flow combinations in parallel in the solution space, providing an initial state for the global search. A Hamiltonian can be constructed. H C :

[0103] (17);

[0104] in, Z i and Z j This represents the Pauli Z operator, whose coefficients are derived from the QUBO model using the standard mapping: =1 / 4 Q ij and d i =-1 / 2 q i -1 / 4∑ j≠i Q ij ,in q i and Q ij These correspond to the elements of matrices q and Q in the QUBO model (4), respectively, while the binary variables... x i ∈{0,1} through x i =(1+Z i ) / 2 is mapped to the Pauli Z operator. QAOA uses a classical optimizer to minimize H under the parameterized variational hypothesis. C The expected value is obtained, thus approximating the ground state energy of the cost Hamiltonian, and the corresponding eigenstate encodes the optimal solution of the original path flow optimization problem.

[0105] First, determine the number of layers in the Quantum Optimization Algorithm (QAOA). s l Then, the parameter vector is initialized using a uniform distribution, expressed as:

[0106] (18);

[0107] in, Cost parameters in the QAOA algorithm No. l The initial value of the layer, It is a uniform probability distribution. Mixed parameters in the QAOA algorithm No. l The initial value of the layer.

[0108] Next is the quantum state evolution process. Based on the parameters ( c , s ), construct the QAOA circuit. At each layer, the cost evolution operator is first applied. :

[0109] (19);

[0110] Among them, H C To compensate for the Hamiltonian, linear terms are implemented using single-qubit Rz gates, while quadratic coupling terms are encoded using Rz gates or an equivalent CNOT-Rz-CNOT structure. Subsequently, a hybrid evolution operator is applied. U M :

[0111] (20);

[0112] in, H M This represents the hybrid Hamiltonian, which is equivalent to applying an Rx rotation gate operation to each qubit. This represents the Pauli-X operator.

[0113] After applying the two operators mentioned above in sequence, there is a layer s l quantum state It can be represented as:

[0114] (twenty one);

[0115] In the formula, Mixed parameters in the algorithm No. l The initial value of the layer, Cost parameters in QAOA No. l The value of the layer.

[0116] The next steps include measuring and evaluating the expected energy. This invention is aimed at computational basis sets. The quantum state is measured. This measurement process will execute... N s Next, a set of samples will be collected. The state... The frequency of occurrence is expressed as N b The formula for estimating the probability is:

[0117] (twenty two);

[0118] Each measurement sample b Each corresponds to a complete path traffic combination scheme, and based on the extracted probability, according to the current variational parameters ( c , s The expected energy value is calculated as follows:

[0119] (twenty three);

[0120] in, This represents the cost value corresponding to this solution. With expected energy... As the energy level is gradually decreased, the probability amplitude of the quantum state gradually converges to a lower-energy pathflow combination, thus approximating the optimal solution. A classic optimizer is used to progressively update the parameters γ and γ. ,For example:

[0121] (twenty four);

[0122] in, or Indicates the learning rate. t Indicates the number of iterations. Indicates expected energy For parameters and The gradient vector is the expected energy change or gradient. The algorithm terminates when the expected energy change or gradient satisfies the convergence condition. After convergence, the final state of the quantum system can be represented as:

[0123] (25);

[0124] After multiple measurements of this state, the path flow combination quantum state The frequency of occurrence satisfies , For path flow combination quantum states The probability amplitude.

[0125] The quantum circuit diagram structure of each QAOA layer is as follows: Figure 3 As shown.

[0126] S4: Fine-tuning based on cubic spline interpolation.

[0127] Quantum optimization algorithms typically output discrete optimization results, while practical engineering applications require continuous flow optimization values. Discrete results cannot directly reflect the continuous optimal flow of the real system. If we rely solely on increasing the number of qubits to improve the discrete accuracy, it will further increase the consumption of computational resources, making it difficult to balance optimization accuracy and computational efficiency.

[0128] Since the discrete results output by QAOA cannot directly reflect the precise values ​​of flow and cost in the real system, the local optimization interval is determined based on the QAOA coarse screening results. ,

[0129] in For QAOA path p Traffic level i The results of the coarse screening, The interval expansion step size. If the coarse screening result is located at the extreme point of the flow rate level (0 or the maximum discrete level), within the interval... Internal selection Ms. 1 discrete sample point; otherwise, select 2. Ms. The nth discrete sample point, the th The number of sample points is ( e =1,2,…,M), where, For discrete sample points, This corresponds to the cost value. Next, natural cubic spline interpolation is used to construct a continuous interpolation function at the sample points to fit the nonlinear relationship between cost and flow rate. The interpolation function is a piecewise cubic polynomial:

[0130] (26);

[0131] in, It is a cubic spline interpolation function. , , , These are the interpolation coefficients. For discrete sample points, To and The flow rate value of the next adjacent discrete sample point. This is the flow rate value. The interpolation condition is... , and For traffic and The corresponding interpolation function value, and For traffic and The corresponding cost value, the first-order continuity condition is that two adjacent cubic spline functions are in... The first derivatives at each point are equal, that is... The second-order continuity condition is Boundary conditions are , Let be the flow rate value of the first discrete sample point, which is the left endpoint of the flow rate interval. Let be the flow rate value of the last discrete sample point, i.e., the right endpoint of the flow rate interval. Afterward, solving for the optimal flow rate transforms into an extremum analysis problem of a piecewise cubic spline function. Since... Constructed from natural cubic splines, its minimum value can be determined by the derivative condition. In each subinterval... superior, It is a cubic polynomial whose first derivative is a quadratic function, and its extreme points satisfy the following: , At most two real roots are generated within each subinterval. Only real roots falling within the corresponding interval are retained as candidate points, and then... It is determined to be a local minimum. Simultaneously, the endpoints of the interval are compared. f 1 and f M The objective function value is used to avoid missing boundary optimal solutions. If multiple local minima exist, their objective function values ​​are compared uniformly, and the smallest one is selected as the global optimum. A candidate set of optimal values ​​is then established. for:

[0132] (27);

[0133] Cubic splines utilize piecewise cubic polynomials to construct a continuous cost-flow function within a low-dimensional candidate interval, extending the discrete optimization results into an analytical continuous form.

[0134] S5: Output the optimization results.

[0135] For different cost-flow bias coefficients l Output the corresponding optimal transmission path, refined optimal traffic, and optimal unit cost, and calculate the fixed transmission volume. F Based on the total transmission cost, a multi-preference transmission optimization strategy is formulated. Simultaneously, the bias coefficient is analyzed. l The impact patterns on optimal paths, costs, and traffic provide a flexible decision-making basis for path selection and traffic configuration in complex networks.

[0136] Based on the same concept, this invention also discloses a single-path traffic transmission optimization system for complex networks, including an acquisition module, a construction module, and a solution module.

[0137] The acquisition module is used to obtain all feasible paths from the source node to the sink node in a directed connected network, the maximum transmission flow of each path, and the bias coefficient used to balance the nonlinear relationship between cost and flow.

[0138] The building module is used to discretize the maximum transmission traffic of each feasible path into multiple traffic levels and the bias coefficient into multiple bias coefficient levels. Logarithmic-level qubits are used to encode the feasible path, traffic level, and bias coefficient level into binary variables, resulting in multiple corresponding encoded variables. The multiple encoded variables are fitted to construct a QUBO model with the optimization objectives of minimizing cost and maximizing traffic. In this model, the coupling terms of multiple encoded variables are equivalently transformed into quadratic terms through auxiliary binary variables.

[0139] The solution module is used to input multiple different encoded variables and auxiliary binary variables into the QUBO model, and solve the QUBO model based on the quantum optimization algorithm to obtain the optimal transmission path, optimal transmission flow and corresponding cost.

[0140] The effects of this invention are as follows:

[0141] Significantly reducing computational complexity and improving the scalability of quantum models: This invention employs logarithmic qubit encoding to encode paths, flow rates, and bias coefficients, transforming the relationship between the number of qubits and network size from linear to logarithmic. Simultaneously, it expands the decision space of the QUBO model from exponential to logarithmic. O (2 j+k Compressed to polynomial order O( j ( k +i This significantly reduces the consumption of quantum computing resources, making quantum optimization algorithms applicable to large-scale complex networks.

[0142] Achieving unified quantum modeling with multiple parameters and solving the problem of redundant calculations under different bias coefficients: This invention embeds path selection, flow level, and cost-flow bias coefficient into the QUBO model in a unified manner. By using auxiliary binary variables, the third-order terms are equivalently transformed into quadratic terms, which meets the modeling requirements of mainstream quantum optimization algorithms and avoids the computational redundancy caused by repeated optimization of multiple parameters under different bias coefficients in traditional methods.

[0143] Quantum-classical co-optimization, balancing global search capability and engineering practicality: A two-stage architecture of quantum coarse screening and classical fine tuning is designed. The superposition and parallelism of the QAOA algorithm are used to achieve efficient screening of global path-flow combinations, solving the local optimum problem of traditional algorithms. Then, the discrete quantum optimization results are fitted into a continuous cost-flow function by cubic spline interpolation to obtain a continuous optimal flow that can be used in engineering, thereby improving the practical application value of the optimization results.

[0144] Adapting to nonlinear cost-flow relationships, the optimization results are more realistic: This invention does not require a unified function modeling of the nonlinear cost-flow relationship. It accurately captures its nonlinear characteristics through interpolation fitting of discrete sample points, solving the problem of collaborative optimization caused by nonlinear coupling. At the same time, the interpolation accuracy can be flexibly adjusted according to engineering needs, taking into account both computational efficiency and fitting accuracy.

[0145] Example

[0146] This invention uses an Internet of Things (IoT) communication network consisting of 122 nodes and 131 edges as a case study to verify its performance. Its structure is as follows: Figure 4 As shown, this network size is sufficient to fully demonstrate the advantages of quantum optimization. Data is transmitted from node 0 to node 100, with a total of 82 transmission paths and a transmission volume of 5000 units. In this numerical example, this invention focuses on the single-source, single-sink case, optimizing under different bias coefficients to obtain the minimum cost of transmitting a specified volume. The feasibility and computational advantages of the proposed method are evaluated using performance metrics such as encoding size, qubit requirement, objective function convergence, and final cost. All paths from node 0 to node 100 are listed in Tables 1-4 below, including all nodes traversed by the path.

[0147] Table 1 Paths 1-20

[0148]

[0149] Table 2 Paths 21-40

[0150]

[0151] Table 3 Paths 41-60

[0152]

[0153] Table 4 Paths 61-82

[0154]

[0155] First, a fitting process is performed, and the decision variables are encoded in binary form as follows. For path encoding, K=7 qubits are used to represent binary indices, resulting in a total of 128 indices. Of these, 82 indices correspond to physically valid paths and are assigned a penalty value of 100. For flow level encoding, to present the results more clearly in the graph and avoid introducing too many qubits (which would make it difficult to distinguish the optimization results), the flow levels are evenly divided into... L =16 levels, using I =4 qubits. For cost-flow coefficient coding, the range is set to [0.25, 0.75], and it is also evenly divided into B=16 levels, using Y = Represented by 4 qubits.

[0156] This invention fits the above-mentioned combination of qubits to QUBO coefficients and then uses QAOA for calculation. Figure 5 The results of the QAOA algorithm are displayed. The horizontal axis represents the binary value of all qubits. For clarity, the horizontal axis is aligned with the bias coefficient. l Path Index r Flow level d The order is converted to decimal form. The auxiliary qubits connecting the bias coefficient qubit, path index qubit, and flux level qubit are not shown in the diagram because they have no actual physical meaning.

[0157] Next, the present invention will... Figure 5 For different bias coefficient values, the optimal path-flow combination is extracted, and the results are as follows: Figure 6 As shown.

[0158] according to Figure 6 The QAOA optimization results should be derived according to formula (2) to derive the traffic corresponding to all bias coefficients, and set the minimum value to 110 and the maximum value to 1000 as the benchmark. Then, calculate the actual traffic value and the cost of 5000 units of transmission traffic, as shown in Table 5.

[0159] Table 5 Actual Flow and Cost

[0160]

[0161] By comparison, it can be seen that when λ=0.6833, the total cost reaches its minimum, path 18 is selected, and the flow rate is 230. It can be observed that several sets of data in Table 5 still have relatively close results. This invention can further refine the flow rate by using cubic spline interpolation, thereby obtaining the exact minimum cost for different λ values. When using cubic spline interpolation to optimize the solution, if the corresponding flow rate obtained by quantum optimization is at the minimum or maximum flow rate level (i.e., 0 or 15), this invention samples the corresponding flow rate value in four different flow rate sub-levels within the sub-interval Γ; otherwise, this invention samples eight flow rate sub-levels. This invention uses different deviation coefficients to fit the spline interpolation function to these sampling points, thereby obtaining a total of 16 optimization results. Specific data are shown in Table 6.

[0162] Table 6 Specific Data

[0163]

[0164] As can be seen, after optimization, the total cost of each bias coefficient is significantly reduced. Furthermore, for a transmission flow of 5000 units, with… l As the coefficient of variation increases, the overall cost shows a downward trend. The minimum overall cost does not occur at the maximum deviation coefficient, but rather at... l When the value is 0.6833, the minimum overall cost is 448.104. This further demonstrates that in this numerical example topology, the cost of traffic transmission is affected by the non-linear relationship between cost and traffic.

[0165] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0166] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A method for optimizing single-path traffic transmission in complex networks, characterized in that, Includes the following steps: Obtain all feasible paths from the source node to the sink node in a directed connected network, the maximum transmission flow of each path, and the bias coefficient used to balance the nonlinear relationship between cost and flow. The maximum transmission traffic of each feasible path is discretized into multiple traffic levels, and the bias coefficient is discretized into multiple bias coefficient levels. Logarithmic-level qubits are used to encode feasible paths, flow rates, and bias coefficients in binary form, resulting in multiple encoded variables. These encoded variables are then fitted to construct a QUBO model with the optimization objectives of minimizing cost and maximizing flow. Auxiliary binary variables are used to convert the coupling terms of the multiple encoded variables in the QUBO model into equivalent quadratic terms. By inputting multiple different encoded variables and auxiliary binary variables into the QUBO model, and solving the QUBO model based on the quantum optimization algorithm, the optimal transmission path, optimal transmission flow, and corresponding cost are obtained.

2. The single-path traffic transmission optimization method for complex networks as described in claim 1, characterized in that, The method employs logarithmic-level qubits to perform binary encoding on feasible paths, flow rates, and bias coefficients, respectively, to obtain multiple corresponding encoded variables, specifically including: use A feasible path for each quantum bit pair p Encode the path to obtain the path-encoded variable. r k ; use Each qubit encodes the flow rate level, resulting in a flow rate encoded variable. d i , L The number of traffic levels; use Each qubit encodes a level of bias coefficient, resulting in a bias coefficient encoded variable. a j , M This represents the number of bias coefficient levels.

3. The single-path traffic transmission optimization method for complex networks as described in claim 2, characterized in that, The process of fitting multiple encoded variables to construct a QUBO model with the optimization objectives of minimizing cost and maximizing traffic specifically includes the following steps: The path-coded variables and flow-coded variables are combined into a binary variable vector. Based on the binary variable vector, the cost function and flow function are mapped to a combination of linear and quadratic terms of the binary variables through least squares fitting, thus obtaining the corresponding fitting expression. The loss function is constructed based on the parameter vector in the fitting expression. The gradient of the loss function with respect to the parameter vector is set to 0. The specific values ​​of the parameter vector are obtained and substituted into the fitting expression to obtain the corresponding QUBO expression. The bias coefficient encoding variables are weighted and summed based on the value range of the bias coefficient to obtain discrete values ​​of the bias coefficient. The discrete values ​​of the bias coefficient are then combined with the QUBO expression to obtain the QUBO model.

4. The single-path traffic transmission optimization method for complex networks as described in claim 3, characterized in that, The process of converting the coupled terms of multiple encoded variables in the QUBO model into equivalent quadratic terms using auxiliary binary variables specifically includes the following steps: Introduce an auxiliary binary variable and add a penalty term with a penalty coefficient to the QUBO model, which forces the auxiliary binary variable to be equal to the product of the two sets of variables in the coupling term; The coupling term is transformed into the standard QUBO form based on auxiliary binary variables and penalty terms.

5. The single-path traffic transmission optimization method for complex networks as described in claim 4, characterized in that, The QUBO model is described in detail below: ； In the formula, r k For the first k Path-encoded variables, d i For the first i Each flow-encoded variable a j For the first j Each bias coefficient encodes a variable. y For auxiliary binary variables, including the first auxiliary variable y jk With the second auxiliary variable y ji , These are the discrete values ​​of the bias coefficient. This is the quadratic form of the QUBO expression for the cost function. This is the quadratic form of the QUBO expression for the flow function. and There are two penalty items.

6. The single-path traffic transmission optimization method for complex networks as described in claim 1, characterized in that, The process of solving the QUBO model using a quantum optimization algorithm to obtain the optimal transmission path, optimal transmission flow, and corresponding cost includes the following steps: The QAOA algorithm is used to solve the QUBO model to obtain the selection result, namely the optimal discrete combination of path-flow-bias coefficient; The selection results are refined by spline interpolation to obtain the optimal transmission path, optimal transmission flow, and corresponding cost.

7. The single-path traffic transmission optimization method for complex networks as described in claim 6, characterized in that, The refinement of the screening results through spline interpolation to obtain the optimal transmission path, optimal transmission traffic, and corresponding cost specifically includes the following steps: Determine a local optimization interval centered on the screening results, and select discrete sample points within the interval; Construct a cubic spline interpolation function that satisfies the continuity of function values, first derivative, and second derivative at discrete sample points; By solving for the minimum value of the interpolation function through extreme value analysis, the optimal transmission path, optimal transmission flow, and corresponding cost can be obtained.

8. The single-path traffic transmission optimization method for complex networks as described in claim 7, characterized in that, The cubic spline interpolation function is as follows: ； In the formula, It is a cubic spline interpolation function. , , , These are the interpolation coefficients. For discrete sample points, To and The flow rate value of the next adjacent discrete sample point. This represents the flow rate.

9. The single-path traffic transmission optimization method for complex networks as described in claim 1, characterized in that, A depth-first search algorithm is used to obtain all feasible paths from the source node to the sink node in a directed connected network.

10. A single-path traffic transmission optimization system for complex networks, characterized in that, include: The acquisition module is used to acquire all feasible paths from the source node to the sink node in a directed connected network, the maximum transmission flow of each path, and the bias coefficient used to balance the nonlinear relationship between cost and flow. A module is constructed to discretize the maximum transmission traffic of each feasible path into multiple traffic levels and the bias coefficient into multiple bias coefficient levels. Logarithmic-level qubits are used to binary encode the feasible path, traffic level, and bias coefficient level respectively, resulting in multiple corresponding encoded variables. The multiple encoded variables are fitted to construct a QUBO model with the optimization objectives of minimizing cost and maximizing traffic. In this model, auxiliary binary variables are used to equivalently transform the coupling terms of multiple encoded variables into quadratic terms. The solution module is used to input multiple different encoded variables and auxiliary binary variables into the QUBO model, and solve the QUBO model based on the quantum optimization algorithm to obtain the optimal transmission path, optimal transmission flow and corresponding cost.

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