The LSTM model [9] was proposed by Hochreiter and Schmidhuber in 1997, which effectively solves the problem of gradient vanishing and gradient explosion encountered when dealing with long sequential data. The structure of LSTM is shown in Figure 1 .

The LSTM model adds three parts to the RNN (Recurrent Neural Network), namely, forgetting gate, input gate and output gate, and each gating unit consists of $\sigma$ and tanh activation function.

1) Forgetting gate: determines which old information is forgotten. $f\left(t\right)$ indicates the importance of the old information, the higher the value, the more important the information.

$f\left(t\right)=\sigma \left(W_f\left[h_{t-1},x_t\right]+b_f\right)$ (1)

2) Input gate: $i_t$ decides which new information is stored. is the output of the input gate, indicating the importance of the new information. ${\tilde{C}}_i$ indicates new information can be stored to the current cell state.

$i_t=\sigma \left(W_i\left[h_{t-1},x_t\right]+b_i\right)$ (2)

${\tilde{C}}_t=\tanh \left(W_c\left[h_{t-1},x_t\right]+b_c\right)$ (3)

At this point, the cell state is updated and the updated state is $C_t$.

3) Output Gate: Determines what information will be output.

$C_t=f_t{C}_{t-1}+i_t{\tilde{C}}_t$ (4)

$h\left(t\right)=\sigma \left(W_o\left[x_t,h_{t-1}\right]+b_o\right)$ (5)

In Equations (1) to (4), W is the network weight coefficient, $h_{t-1}$ is the output of the previous time step, $h_t$ is the output of the current time, and b is the bias of the activation value.

The average absolute error (MAE), root mean square error (RMSE), average absolute percentage error (MAPE) and Symmetric Mean Absolute Percentage Error (SMAPE) are usually used to evaluate the prediction effect of the model, and the lower the value of the indexes, the more stable the model is and the more accurate the prediction results are. The formulas are as follows:

$\text{MAE}=\frac{1}{m}{\displaystyle \underset{t=1}{\overset{m}{\sum }}\left|\left(h_t-{\hat{h}}_t\right)\right|}$ (6)

$\text{RMSE}=\sqrt{\frac{1}{m}{\displaystyle \underset{t=1}{\overset{m}{\sum }}{\left(h_t-{\hat{h}}_t\right)}^2}}$ (7)

$\text{MAPE}=\frac{100\%}{m}{\displaystyle \underset{t=1}{\overset{m}{\sum }}\left|\frac{{\hat{h}}_t-h_t}{h_t}\right|}$ (8)

$\text{SMAPE}=\frac{100\%}{m}{\displaystyle \underset{t=1}{\overset{m}{\sum }}\frac{\left|h_t-{\hat{h}}_t\right|}{\left|h_t\right|+\left|{\hat{h}}_t\right|}}$ (9)

where m, $h_t$, and ${\hat{h}}_t$ are the total number of samples, measured values, and predicted values, respectively. MAE measures the average size of the absolute value of the difference between the predicted and actual values; RMSE is the square root of the mean of the squares of the difference between the predicted and actual values; MAPE is the average percentage of the absolute value of the ratio of the prediction error to the actual value; SMAPE reduces the problem of numerical inflation that may occur with MAPE by taking the absolute value of the predicted value and the actual value and averaging them.

Metropolis et al. [10] proposed the idea of Simulated Annealing (SA) in 1953, which has since been widely used in combinatorial optimization [11] problems. SA, as a heuristic global optimization algorithm, simulates the physical process of atoms rearranging to reach thermodynamic equilibrium during annealing of solid materials. Although SA can handle complex optimization problems, its convergence speed and and solution quality depend on the parameter settings, and the temperature of the object is directly proportional to the energy, at lower temperatures, according to the Metropolis criterion, it only accepts new solutions that are smaller than the current solution, or when the temperature decreases at too fast a rate, it may skip many potential globally optimal solutions, and thus easily causes the system to stay at the local optimal solution.

With the development of quantum technology, Quantum Annealing (QA) algorithm [12] based on quantum tunneling effect is proposed. The so-called quantum tunneling effect, i.e., the quantum leap gives the quantum the ability to penetrate the potential barrier higher than its own energy. Simulated annealing algorithm and quantum annealing algorithm over the potential barrier are shown in Figure 2 , the particle reaches the local optimum at point A, the simulated annealing algorithm needs to go over the potential barrier in order to achieve the global optimum, and the quantum annealing algorithm only needs to reach the global optimum point B directly through the quantum tunneling effect.

The process of realizing quantum annealing requires 6 steps:

Step 1: Construct the evaluation function $H_q=H_{pot}+H_{kin}$ of the quantum system, where $H_{pot}$ is the potential energy to represent the objective function of the optimization problem and $H_{kin}$ is the kinetic energy;

Step 2: Set the initial temperature T, the transverse magnetic field strength Γ, and the maximum number of iterations Steps. Select an initial state x randomly at temperature T and calculate $H_q\left(x\right)$;

Step 3: Generate a new state x' by randomly perturbing state x and calculate its $H_q\left(x\prime \right)$;

Step 4: Calculate $\Delta H_q=H_q\left(x\right)-H_q\left(x\prime \right)$. If $\Delta H_q>0$, or $\Delta H_q<0$ and ${\text{e}}^{-\frac{\Delta H_q}{T}}<\text{random}\left(0,1\right)$, the new state x' is accepted, otherwise, return to step 3;

Step 5: Perform a de-tempering and magnetic field attenuation operation;

Step 6: Determine whether the maximum number of iterations or the termination condition is reached, if so terminate, otherwise return to step 3.

According to the above steps, the flowchart of quantum annealing algorithm is shown in Figure 3 :

After a data stream is generated, there are multiple paths between its source and destination nodes in the data center, and choosing the appropriate path for forwarding can effectively maintain network load balancing. In this paper, we design the optimal forwarding path policy that satisfies the constraints through the current and predicted load balancing degree.

Construct the data center network as a directed graph $G=\left(S,L\right)$, with S being the set of all nodes in the network and L being the set of all links in the network. Denote the path between the data flow source node s and the destination node d as P_s,d.

In a data center network, choosing a path each time to forward with the maximum available bandwidth between source and destination nodes can effectively alleviate and prevent load imbalance problems. Get the switch port statistics to get the number of bytes sent and received by the port and calculate the transmission rate $spee{d}_{i,j}$ of the port based on the last recorded number of bytes transmitted:

$spee{d}_{i,j}=\frac{stat{s}_{i,j,t}-stat{s}_{i,j,t-1}}{pd}$ (10)

In Equation (10), i and j respectively denote the source and destination switches, stats denotes the number of bytes transmitted on the port, and pd denotes the time interval between two recordings.

Then the bandwidth $b{w}_l$ of link l is denoted as the minimum of the two ports connected, and the available bandwidth $free\text{\_}bw$ is the difference between the link capacity $C_l$ and the used bandwidth, and the minimum available bandwidth of all the links in the path is denoted as the available bandwidth of the path:

$b{w}_l=\min \left(spee{d}_{i,j},spee{d}_{j,i}\right)$ (11)

$free\text{\_}b{w}_l=C_l-b{w}_l$ (12)

$A=\min \left(free\text{\_}b{w}_{l_1},free\text{\_}b{w}_{l_2},\cdots ,free\text{\_}b{w}_{l_n}\right)$ (13)

A link is often part of multiple paths, so the available bandwidth of different links in a path varies. To avoid overloading certain links and improve transmission efficiency, the load balancing degree of a link is considered when selecting a forwarding path. In this paper, the bandwidth equalization degree of a path is expressed as the difference between the minimum and maximum values of the bandwidth between all nodes in the path:

$B=\frac{\min \left(free\text{\_}b{w}_{l_1},free\text{\_}b{w}_{l_2},\cdots ,free\text{\_}b{w}_{l_n}\right)}{\max \left(free\text{\_}b{w}_{l_1},free\text{\_}b{w}_{l_2},\cdots ,free\text{\_}b{w}_{l_n}\right)}$ (14)

Combining the available bandwidth and bandwidth equalization of the above paths with the LSTM prediction of the available traffic and bandwidth equalization, the transmission cost of the paths is used as a weight for path selection. In addition, in order to guarantee the transmission of the data flow, one path is selected at a time and only one path is selected, and there are no loops in the selected paths, so that the objective function definition is obtained:

$\begin{array}{l}\min \frac{1}{A+B+A_{pre}+B_{pre}}\\ \text{s}\text{.t}.\{\begin{array}{l}{\displaystyle \underset{r=1}{\overset{n}{\sum }}{x}_r\le 1}\hfill \\ P=p\hfill \end{array}\end{array}$ (15)

In Equation (15), n is the number of all forwarding paths calculated from the source destination address.

The optimal path is solved by quantum annealing algorithm and the solution process is shown in Algorithm 1 :

Algorithm 1. Optimal path selection based on LSTM and QA.