We thank Yulin Chi, Gang Xi Wang, Xinying Li, Xin Yi, and Fei Wang for their insightful discussions.

QTSMixer methodology
Let <img alt="Matrix equation \(X_{s \times l \times c} \in \mathbb{R}^{s \times l \times c}\); mathematical symbol." src="/files/ftp_upload/68229/68229eq01.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 1"> be a multivariate time series of length sl and number of channels c. The multivariate forecasting task is defined as predicting future values <img alt="Mathematical formula Y in R space; concept: matrix representation in linear algebra." src="/files/ftp_upload/68229/68229eq02.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 2"> given some history <img alt="Static equilibrium, ΣFx=0, diagram, mechanical forces analysis, engineering study, vector resolution." src="/files/ftp_upload/68229/68229eq03.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 3"> by a forecasting model <img alt="Static equilibrium; ΣFx=0; diagram; mechanics analysis; force vectors; concise educational use." src="/files/ftp_upload/68229/68229eq04.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 4">, which can be formulated as
<img alt="Mathematical model equation showing statistical prediction for scientific data analysis." src="/files/ftp_upload/68229/68229eq05.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 5">
Where <img alt="Static equilibrium formula ΣFx=0; equation diagram; physics calculation concept." src="/files/ftp_upload/68229/68229eq06.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 6"> is the forecast sequence length and <img alt="Mathematical symbol for statistical estimator in data analysis, relevant to predictive modeling." src="/files/ftp_upload/68229/68229eq07.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 7"> denotes the prediction value. In QTSMixer, let XL×C denote the original multivariate time series of length L and the number of channels c. We input the multivariate time series by minibatch <img alt="Mathematical notation, matrix dimensions, formula showing tensor dimensions in multivariable calculus." src="/files/ftp_upload/68229/68229eq20.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 20"> , where sl ≤ L and b is the batch size. As shown in Figure 1, the QTSMixer model consists of four components: normalization, patch embedding, mixing layers, and forecasting. In the training phase, the loss of the model is calculated by the Mean Square Error (MSE) loss function <img alt="Static equilibrium diagram; ΣFx=0; beam balance; forces, torque; educational physics concept." src="/files/ftp_upload/68229/68229eq21.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 21"> to achieve the best fit. The QTSMixer can be characterized as
<img alt="Optimization equation for machine learning, argmin, loss function, mathematical formula." src="/files/ftp_upload/68229/68229eq08.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 8">
Where θ is the trainable parameter vector and is updated by stochastic gradient descent.
<img alt="Time series prediction diagram; process: normalization, patch embedding, mixing layer, forecasting." class="xfigimg" src="/files/ftp_upload/68229/68229fig1.jpg" data-altupdatedat="1788304762000" data-alt="Figure 1"> 
Figure 1: The high-level architecture of QTSMixer. Here we divide QTSMixer into four components: normalization, patch embedding, mixing layers, and forecasting. <a href="https://www.jove.com/files/ftp_upload/68229/68229fig1large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
Next, we will detail the model components of QTSMixer.
Normalization
The forward propagation process of QTSMixer starts with receiving the input time series data <img alt="Equation, symbol notation for matrix dimensions in tensor algebra, educational diagram." src="/files/ftp_upload/68229/68229eq09.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 9"> . First, the input data are normalized and standardized by the PatchTSMixer StandardScaler8 or other scalers to eliminate the dimensional differences between features and ensure the stability of data performance during training.
Patch embedding
First, the standardized time series data are divided into multiple fixed-length segments (patches) through the patching module. This model design is inspired by Ekambaram et al.8, and the ability to capture local features is improved through the division of patches. Through patch division, QTSMixer can effectively deal with complex time series data patterns, reduce the number of model input tokens, and achieve better results in a shorter training time. The minibatch <img alt="Matrix equation, \(X^B_{b \times sl \times c}\), illustrating tensor dimensions, scientific formula." src="/files/ftp_upload/68229/68229eq10.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 10"> is reshaped into <img alt="Static equilibrium equation, \(X^P_{b \times n \times p \times l \times c} \in \mathbb{R}^{b \times n \times p \times l \times c}\), formula." src="/files/ftp_upload/68229/68229eq11.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 11">, where n is the number of patches and pl denotes the patch length. Let s denote the stride of patching, then <img alt="Equation for pipeline stages; n=|sl-pl|/s+1; diagram; educational formula." src="/files/ftp_upload/68229/68229eq12.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 12"> .
To increase the expressive power of the model, each patch is mapped to a higher-dimensional space through a linear layer. Then, the data are reshaped into <img alt="Equation for data tensor in neural network, symbol: \(X = b \times n \times hf \times c\)" src="/files/ftp_upload/68229/68229eq13.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 13"> by linear transformation <img alt="Functional mapping notation, ℝ^pl ↦ ℝ^hf, in mathematical diagram." src="/files/ftp_upload/68229/68229eq14.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 14">, where hf is the number of hidden features. The weight matrix has a dimension of pl × hf .
Mixing layers
In this part, the patch, feature, and channel dimensions are mixed through multiple mixing layers. Based on the dimension under focus in each mixer layer, the input is first permuted accordingly to learn correlation along the focused dimension. Then, the information is extracted by layer normalization, MLP, QNN, and Gated Attention (GA) mechanism to extract more comprehensive feature information.
In the patch mixing layer, the input is first reshaped to <img alt="Mathematical expression, XP b×h f×c×n; educational research, formula analysis." src="/files/ftp_upload/68229/68229eq15.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 15"> , in which we focus on the patch dimension (i.e., the last dimension); then, the layer normalization is performed. After that, this module employs a shared MLP (weight dimension n × n) to learn the correlation between different patches. Meanwhile, the QNN is realized by quantum re-uploading technology20 to enhance the expression ability of the quantum layer. The quantum circuit in QNN has a total of ql layers, and the ith layer contains an encoding circuit Ux and an ansatz <img alt="Static equilibrium equation Uθᵢ, structural analysis diagram, balance forces, mechanical system." src="/files/ftp_upload/68229/68229eq22.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 22">, where θi is the trainable parameter vector and i = 1, 2, ..., ql. Following a weighted addition operation, the model produces a hybrid structure of MLP and QNN, formulated as <img alt="Neural network function formula MLP(x_i) + α_i × QNN(x_i); symbolic representation." src="/files/ftp_upload/68229/68229eq16.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 16"> for the ith component of the patch dimension, to effectively integrate classical and quantum feature extraction capabilities. Note that here αi is a trainable parameter with an initial value of 0, which indicates how strongly the quantum behavior is introduced. Finally, the GA8 probabilistically upscales the dominant features and downscales the unimportant features. The feature mixing layer and the channel mixing layer work in a similar way.
After multiple mixing layers, the data are processed as <img alt="Mathematical equation with exponents, variables X, P, describing transformation in data analysis." src="/files/ftp_upload/68229/68229eq17.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 17">.
Forecasting
In the forecasting layer, the data are first reshaped to <img alt="Static equilibrium equation X^P_{b×c×n×h×f}, mathematical symbol for educational analysis." src="/files/ftp_upload/68229/68229eq18.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 18">. The future values are predicted through a classical prediction head8—the encoded features are flattened and fed into a classical linear layer with dropout to generate a prediction <img alt="Static equilibrium equation: Ŷb×Fl×c; mathematical formula in physics.

" src="/files/ftp_upload/68229/68229eq19.jpg" style="margin: auto;" data-altupdatedat="1788304762000" data-alt="Equation 19"> of future values.
Experimental results
We conducted a detailed empirical analysis on the real-world dataset, i.e., the Long-Term Network Traffic Forecasting (LTNTF) dataset21, and three other popular public datasets. The first experiment is based on the LTNTF dataset21, which provides hourly traffic data and holiday information for three cities A, B, and C, in the real-world network from January 1, 2017, to February 20, 2019. The task is to use historical data to build an appropriate model to predict the hourly traffic values for each city for the next 95 days. The uniqueness of the dataset lies in its authenticity and long-term nature, offering detailed hourly network traffic data covering multiple cities over more than 800 days. The experimental results are summarized in Table 1, Figure 2, and Figure 3. To reproduce the experiment, refer to README.md in Experiment_1_LTNFT in Supplemental File 1.
Table 1: Performance comparison between QTSMixer and TSMixer in the LTNTF dataset. Here the MAPE and RMSE are used as the main evaluation metrics, and lower values indicate better performance. <a href="https://www.jove.com/files/ftp_upload/68229/68229_Table%201.xlsx" target="_blank">Please click here to download this Table.</a>
<img alt="Time-series data analysis graph showing multiple city samples, predicting patterns and trends." class="xfigimg" src="/files/ftp_upload/68229/68229fig2.jpg" data-altupdatedat="1788304762000" data-alt="Figure 2"> 
Figure 2: Comparison of prediction results of TSMixer and QTSMixer. Here we randomly select three samples from three cities and give the visual comparison between the real values and the predicted values of TSMixer and QTSMixer. <a href="https://www.jove.com/files/ftp_upload/68229/68229fig2large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Time series analysis graph; comparisons of city temperature trends; includes historical data and forecasts." class="xfigimg" src="/files/ftp_upload/68229/68229fig3.jpg" data-altupdatedat="1788304762000" data-alt="Figure 3"> 
Figure 3: Comparison of prediction results of a special case of TSMixer and QTSMixer. Here the samples with the best performances of QTSMixer are selected. <a href="https://www.jove.com/files/ftp_upload/68229/68229fig3large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
The second experiment is based on the BasicTS+(Basic Time Series)22 framework, which is a benchmark library and toolbox for time series forecasting. It supports a variety of tasks and datasets, such as spatio-temporal forecasting and long series forecasting, and covers statistical models, machine learning models, deep learning models, and other algorithms. BasicTS+ provides a fair and comprehensive platform for the replication and comparison of popular deep learning models through a unified and standardized process. The BasicTS benchmark comprises seven carefully curated multivariate time series (MTS) datasets spanning diverse real-world domains to enable rigorous evaluation of forecasting models. The collection includes traffic datasets (PEMS04/08 for flow monitoring), energy consumption records (Electricity/Etth1/Ettm1), economic indicators (Exchange-Rate), and environmental measurements (Beijing Air Quality). The experimental results are shown in Table 2. To reproduce the experiment, refer to README.md in Experiment_2_Basicts in Supplemental File 1.
Table 2: Performance comparisons among QTSMixer and other models on various datasets of BasicTS+ framework. Here MAE, WAPE and RMSE are used as evaluation metrics, and lower values indicate better performance. The models with similar parameter counts are grouped. <a href="https://www.jove.com/files/ftp_upload/68229/68229_Table%202.xlsx" target="_blank">Please click here to download this Table.</a>
The third experiment involves a comparative analysis between the quantum simulator and a real quantum computer. This experiment compares the performance of Qiskit and the DQ framework by evaluating their respective implementations of QTSMixer. The accuracy and execution efficiency of both frameworks in forecasting tasks were assessed. The superconducting quantum computer, ibm_brisbane, is used in this experiment. The experimental results are shown in Figure 4 and Table 3. To reproduce the experiment, refer to README.md in Experiment_3_Qiskit in Supplemental File 1.
<img alt="Time series analysis, actual vs. predicted values graph, showing prediction accuracy comparison." class="xfigimg" src="/files/ftp_upload/68229/68229fig4.jpg" data-altupdatedat="1788304762000" data-alt="Figure 4"> 
Figure 4: Comparison of prediction results of Qiskit (on quantum hardware) and DQ (on the simulator). Comparison between actual values and predictions, with predictions derived from Qiskit and DQ experiments. <a href="https://www.jove.com/files/ftp_upload/68229/68229fig4large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
Table 3: Performance comparison of QTSMixer between quantum hardware and simulation experiments. Here, the MAPE and RMSE are used as the main evaluation metrics, and lower values indicate better performance. <a href="https://www.jove.com/files/ftp_upload/68229/68229_Table%203.xlsx" target="_blank">Please click here to download this Table.</a>

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The authors declare that they have no competing financial interests.

In this paper, we proposed QTSMixer for long-term NTP, which is a novel quantum-classical hybrid approach with improvements in accuracy and cross-domain adaptability. This study not only expands the application potential of quantum computing in time series prediction but also points out the direction for its future deployment on real quantum hardware. However, the current quantum-classical hybrid architecture exhibits limited capability in processing exogenous variables. Future improvements could incorporate a channel-mixer block to enhance cross-variable information extraction while maintaining quantum efficiency.
Here, the QTSMixer model based on QNN and MLP has higher prediction accuracy than TSMixer when dealing with NTP problems. To the best of our knowledge, QTSMixer is the first quantum-classical hybrid model for long-term NTP.
Quantum-classical hybrid architecture: 
QTSMixer employs a hybrid quantum-classical architecture. In QTSMixer, quantum processing layers are embedded within traditional time series models. These quantum layers are more adept at extracting features related to periodic signals and long-term dependencies in the data.
Cross-domain application potential: 
The design of QTSMixer is not limited to specific types of time series data. Its structure and methods are highly generalizable, making it easily adaptable to different domains, such as energy consumption forecasting, stock market trend analysis, and long-term network traffic forecasting, demonstrating broad cross-domain application potential.
The QTSMixer model adopts a hybrid structure of Multilayer Perceptron (MLP) and Quantum Neural Network (QNN), represented as MLP(x) + α×QNN(x), to effectively integrate classical and quantum feature extraction capabilities. Here, α is a trainable parameter that starts from zero. As shown in Figure 5 and Figure 6, it can be observed that as training progresses, the distribution of all elements of the tensor gradually spreads from being concentrated near zero. This indicates that, to minimize the loss function, the stochastic gradient descent algorithm gradually integrates the features outputted by the QNN into the classical network by adjusting the values of α.
<img alt="Tensor parameter distribution chart showing patch_mixer.alpha variation over training epochs." class="xfigimg" src="/files/ftp_upload/68229/68229fig5.jpg" data-altupdatedat="1788304762000" data-alt="Figure 5"> 
Figure 5: The changes in α in patch mixing module. Changes in the distribution of the trainable parameter in the patch mixing module as training progresses. <a href="https://www.jove.com/files/ftp_upload/68229/68229fig5large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Tensor distribution chart, feature_mixer.alpha vs. epochs, parameter analysis, neural network training." class="xfigimg" src="/files/ftp_upload/68229/68229fig6.jpg" data-altupdatedat="1788304762000" data-alt="Figure 6"> 
Figure 6: The changes in α in feature mixing module. Changes in the distribution of the trainable parameter α in the feature mixing module as training progresses. <a href="https://www.jove.com/files/ftp_upload/68229/68229fig6large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
In future research, we plan to expand the application and enhance the performance of QTSMixer through several avenues. We aim to apply QTSMixer to a broader range of long-term multivariate time series forecasting scenarios, such as financial market analysis and weather prediction. To improve the model's applicability across various domains, we will investigate cross-domain learning techniques to enable QTSMixer to generalize more effectively to new datasets. Additionally, we will explore methods to optimize the quantum-classical hybrid architecture to increase the model's predictive accuracy and computational efficiency. With these efforts, we anticipate that QTSMixer will play a more significant role in practical applications and offer innovative solutions in the field of time series forecasting.

Network Traffic Prediction (NTP)1,2,3, as an essential application of time series analysis4,5, involves analyzing historical network traffic data to identify patterns and trends, thereby forecasting future traffic changes. It is crucial for network management and optimization, as it can assist network operators in resource allocation, fault diagnosis, security monitoring, and quality of service assurance. The NTP problem has been well studied by statistical models and machine learning methods3. Recently, architectures based on Multi-Layer Perception Mixer (MLP-Mixer)6,7 are becoming popular for NTP due to their powerful capability to capture multivariate time series dependencies. Among them, TSMixer8 stands out for its exceptional speed and lightweight design and has attained State-Of-The-Art (SOTA) performance on benchmark datasets9. TSMixer can better capture the time dependence and the correlation between features through the hybrid operation of time and feature dimensions. However, TSMixer has the following two disadvantages. First, it is not suitable for dealing with periodic signals. Periodic signals are stable, resistant to interference, and easy to analyze, making them highly valuable in communication, power systems, and control systems. Since TSMixer is based on MLP whose mathematical prototype is an aperiodic function, TSMixer is suitable for processing aperiodic signals but does not fit well when processing periodic signals. Second, it does not perform well when making long-term predictions as it is hampered by current computer memory limitations and the accumulation of errors in predicting. At present, most of the models have poor performance in dealing with long-term time series prediction. This is an urgent issue that needs to be solved in the field of time series prediction. Although TSMixer has achieved good results, there is still room for improvement. Therefore, it is of great significance to seek new strategies to avoid these drawbacks.
Quantum computing10 is a new computing paradigm based on quantum mechanics laws, which can process massive amounts of data both efficiently and securely. Quantum Neural Networks (QNNs)11,12,13 are a kind of computing model that combines quantum computing and classical neural networks, which have a wide range of applications in image processing, speech recognition, and disease prediction. They exploit the properties of quantum computing, such as superposition and entanglement, to process information and perform machine learning tasks, leading to better simulations, faster computation, or quantum advantage in terms of network performance. Recently, some approaches have been investigated in the QNN realm for time series forecasting14,15,16,17,18,19. Quantum time series models can surpass traditional models in terms of prediction accuracy and computational efficiency, especially when dealing with complex multi-dimensional data. On the one hand, QNN can be expressed as a Fourier series, which is a periodic function and suitable for processing periodic signals; on the other hand, some applications of QNN in time series have been successful. Therefore, it is of great interest to explore the role of QNN in NTP, especially long-term NTP.

<table><tbody><tr><td>DeepQuantum</td><td>TuringQ Co., Ltd.</td><td>https://deepquantum.turingq.com/</td><td>An efficient programming framework for quantum machine learning and optical quantum computing developed by TuringQ.</td></tr><tr><td>Qiskit</td><td>IBM</td><td>https://www.ibm.com/quantum/qiskit</td><td>An open-source SDK for working with quantum computers at the level of extended quantum circuits, operators, and primitives.</td></tr><tr><td>ibm_brisbane</td><td>IBM</td><td>https://quantum.ibm.com/</td><td>The superconducting quantum computer in the IBM Quantum Eagle family.</td></tr><tr><td>LTNTF dataset</td><td>China Mobile</td><td>https://jiutian.10086.cn/open/#/dataset/710012?platform=OpenInnovation</td><td>The dataset comes from the Jiutian AI platform of China Mobile.</td></tr><tr><td>python3.10</td><td>Python Software Foundation</td><td>https://www.python.org/downloads/release/python-3100/</td><td></td></tr><tr><td>quantum re-uploading technology</td><td>P&amp;eacute;rez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J. I. Data re-uploading for a universal quantum classifier. Quantum. 4 226 (2019).</td><td>https://doi.org/10.22331/q-2020-02-06-226</td><td></td></tr><tr><td>BasicTS+ framework</td><td>Shao, Z. et al. Exploring progress in multivariate time series forecasting: Comprehensive benchmarking and heterogeneity analysis. IEEE Transactions on Knowledge and Data Engineering. 37 291-305 (2023).</td><td>arXiv:2310.06119v2</td><td></td></tr><tr><td>GPU A100</td><td>NVIDIA</td><td></td><td>80G GPU</td></tr><tr><td>Transformers</td><td>Hugging Face</td><td>https://huggingface.co/</td><td>The platform where the machine learning community collaborates on models, datasets, and applications.</td></tr></tbody></table>

Network traffic prediction, a critical application of time series analysis, is essential for network management and optimization. Traditional statistical models and machine learning methods have been employed for network traffic prediction, where recent advancements in multi-layer perception mixer architectures, particularly TSMixer, have achieved state-of-the-art performance. However, TSMixer faces challenges with periodic signals and long-term predictions. To address these limitations, we propose a Quantum TSMixer (QTSMixer) model, a hybrid quantum-classical approach that leverages quantum neural networks for enhanced feature extraction related to periodic signals and long-term dependencies. By introducing trainable parameters to control the strength of the quantum components, the hybrid structure of multi-layer perception and quantum neural network is generated. QTSMixer's cross-domain application potential and practical application capability are demonstrated through empirical analysis on real-world datasets, in which QTSMixer outperforms TSMixer by 6.72% in the long-term network traffic prediction dataset. In the future, the development of QTSMixer can be further explored in other fields, such as financial market analysis and weather prediction.

We conducted a detailed empirical analysis on some datasets. The first experiment is based on the LTNTF dataset. The three cities exhibited distinct data distribution patterns: City A showed a significant low point in traffic flow at the beginning of the year; City B maintained relatively high values during the same period, while City C displayed more balanced distribution throughout the year, see Supplemental Figure&#160;S1, Supplemental Figure&#160;S2, and Supplemental Figure&#160;S3 for details. The experimental results are summarized in Table 1, in which the quantum circuit module is implemented by the DeepQuantum (DQ) framework. The model prediction is denormalized, and the Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE) are used as the main evaluation metrics. Compared with the classical model, the QTSMixer model shows stronger generalization ability on the test set and outperforms TSMixer by 6.72%. In Figure 2 and Figure 3, the visual comparison between the predicted values and the real values further verifies the robustness and accuracy of the model.
The second experiment is based on the BasicTS+(Basic Time Series)22 framework. The experimental results are shown in Table 2, where the Mean Absolute Error (MAE), Weighted Absolute Percentage Error (WAPE), and RMSE are used as evaluation metrics and the quantum circuit modules are implemented by the DQ framework. The results of QTSMixer and TSMixer are compared to other models as the last two columns of Table 2, where the other model results are from Table VI in Shao et al. 22. In Table 2, we compare models with similar parameter counts grouped together. Notably, the larger transformer-based architectures, which naturally contain more parameters, typically achieve superior prediction metrics. The experiment verifies the effectiveness of QTSMixer in multivariate time series prediction under the framework of BasicTS+, especially on high-dimensional data sets. The results indicate that our proposed model outperforms TSMixer in most of the experiments. While QTSMixer demonstrates strong performance on the PEMS datasets, ETTh1 dataset, and weather forecasting tasks, its predictive capabilities may be less optimal for certain other datasets. Future work will focus on improving the spatiotemporal adaptability of the model to reduce the error.
The third experiment involves a comparative analysis between the quantum simulator and a real quantum computer by performing the LTNTF dataset. Figure 4 presents a comparison between the actual values and the predictions made by Qiskit (on quantum hardware) and DQ (on the simulator). The blue curve represents the actual values; the red dashed line represents the predictions from Qiskit (on quantum hardware); and the green dashed line represents the predictions from DQ (on the simulator). It is evident from the graph that the prediction trends of both models are consistent. Based on the experimental outputs, we also computed the RMSE and MAPE metrics, as shown in Table 3. Specifically, Qiskit (on quantum hardware) had an RMSE of 2.0006 and a MAPE of 8.8724%, while DQ (on the simulator) had an RMSE of 1.9947 and a MAPE of 8.7289%. These results indicate that DQ (on the simulator) slightly outperforms Qiskit (on quantum hardware) in terms of predictive accuracy for this task. The simulator&#39;s superior accuracy stems from idealized zero-noise assumptions, while the observed precision discrepancies in quantum hardware primarily originate from: (1) limitations in qubit coherence times (T1/T2); (2) cumulative effects of two-qubit gate errors (~0.2% per gate); and (3) readout noise-induced final-state measurement distortion. Despite exhibiting relatively lower precision, the quantum hardware achieves computationally scalable regimes inaccessible to classical simulation, thereby realizing the algorithm&#39;s scalability. In addition, the QNN used in the proposed protocol only needs O(n) CNOT gates, where n denotes the number of qubits in the quantum circuit, the circuit is relatively shallow, and the complexity is low. It can be implemented on noisy intermediate-scale quantum (NISQ)23 devices while maintaining low computational overhead. Hence, while combining classical MLPs with QNNs does introduce additional computational complexity, the benefits in terms of adaptability and performance improvement justify this burden, especially when considering the potential for future advancements in quantum computing.
DATA AVAILABILITY: 
Datasets: 
LTNFT Dataset:&#160;https://doi.org/10.12448/3fp4-w20j 
BasicTS+ Framework: BasicTS/datasets at master &#183; GestaltCogTeam/BasicTS &#183; GitHub
Paper codes (see <a href="https://www.jove.com/files/ftp_upload/68229/68229_Supp File-1.zip" target="_blank">Supplemental File 1</a>): 
Code 1: Experiment_1_LTNFT 
Code 2: Experiment_2_Basicts 
Code 3: Experiment_3_Qiskit
Supplemental Figure S1: The network traffic data of City A. The original time series diagram (top) shows the overall trend of network traffic in city A, where the blue, green, and red areas correspond to the training set, validation set, and test set, respectively. The normalized time series plot (middle panel) shows the data after normalization, with the mean normalized to 0. The detailed plot of the last 2 months of data (below) further magnifies the performance of the normalized data on a short time scale, highlighting the periodic fluctuations of the traffic. <a href="https://www.jove.com/files/ftp_upload/68229/Figure S1.pdf" target="_blank">Please click here to download this File.</a>
Supplemental Figure S2: The network traffic data of City B. The original time series diagram (top) shows the overall trend of network traffic in city B, where the blue, green, and red areas correspond to the training set, validation set, and test set, respectively. The normalized time series plot (middle panel) shows the data after normalization, with the mean normalized to 0. The detailed plot of the last 2 months of data (below) further magnifies the performance of the normalized data on a short time scale, highlighting the periodic fluctuations of the traffic. <a href="https://www.jove.com/files/ftp_upload/68229/Figure S2.pdf" target="_blank">Please click here to download this File.</a>
Supplemental Figure S3: The network traffic data of City C. The original time series diagram (top) shows the overall trend of network traffic in city C, where the blue, green, and red areas correspond to the training set, validation set, and test set, respectively. The normalized time series plot (middle panel) shows the data after normalization, with the mean normalized to 0. The detailed plot of the last 2 months of data (below) further magnifies the performance of the normalized data on a short time scale, highlighting the periodic fluctuations of the traffic. <a href="https://www.jove.com/files/ftp_upload/68229/Figure S3.pdf" target="_blank">Please click here to download this File.</a>

QTSMixer, a hybrid quantum-classical model, enhances network traffic prediction by addressing TSMixer's limitations in periodic signals and long-term forecasts. It introduces quantum neural network components controlled by trainable parameters , outperforming TSMixer by 6.72% in long-term predictions on real-world data.

a-quantum-classical-hybrid-model-for-long-term-network-traffic

A Quantum-Classical Hybrid Model for Long-term Network Traffic Prediction

Network traffic prediction, a critical application of time series analysis, is essential for network management and optimization. Traditional statistical ...

Braulio Misael Villegas Martinez

The collection will cover a range of topics, from fundamental quantum mechanics to applied quantum technologies, offering a bridge between foundational knowledge and practical applications. We welcome contributions that delve into theoretical foundations, experimental implementations, and technological advancements in quantum information processing. The collection covers various subjects, including but not limited to:&nbsp;Quantum Computing:&nbsp;<ul><li class="ql-align-justify">Quantum computer architectures and implementations&nbsp;</li><li class="ql-align-justify">Quantum software and programming languages&nbsp;</li><li class="ql-align-justify">Applications of quantum computing&nbsp;&nbsp;</li></ul>Quantum Information Theory:&nbsp;<ul><li class="ql-align-justify">Foundations of quantum mechanics and its implications for information processing&nbsp;</li></ul>Quantum Optics and Photonics:&nbsp;<ul><li class="ql-align-justify">Quantum optics experiments and technologies&nbsp;</li></ul>

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A Quantum-Classical Hybrid Model for Long-term Network Traffic Prediction