# Sea Drift Trajectory Prediction Based on Quantum Convolutional Long Short-Term Memory Model
## 3 Methods
### QCNN-LSTM Model Architecture  
The QCNN-LSTM model requires three pairs of quantum convolutional and quantum pooling layers to accept quantum-encoded data, represented by a single quantum circuit composed of 6 qubits, with each qubit representing an input feature. A parameterized quantum circuit layer (PQC) is added after the QCNN layer to obtain the output as an MLP. The \(Z\)-axis transformation is used to measure the quantum state of the qubit. A differential operator is used to explain how to measure the gradient of the quantum circuit. The differential operator serves as the quantum gradient descent function, which performs differentiation through forward and backward propagation of the quantum circuit. The Adam optimizer is used during the optimization process of the QCNN layer to estimate the parameters through backpropagation and minimize the mean squared error. The convolutional part of the quantum convolutional layer uses the \"LeakyReLU\" activation function to evaluate the output from the convolutional operator, the bias term, and the feature map of the continuous layer. The following equation gives the evaluation of the output feature map of the convolutional layer:  
\[x_{j}^{l}=lrelu\left(x_{j}^{l}\right) \tag{6}\]  
\[z_{j}^{l}=\sum_{i}x_{i}^{l-1}\odot k_{ij}^{l}+b_{j}^{l} \tag{7}\]  
The input of the convolutional filter is \(x_{i}^{l-1}\), the output \(x_{t}^{l}\), \(k\) is the convolution kernel, b is the paranoia term, and \(lrelu()\) stands for the \"LeakyReLU\" function. \(\odot\) denotes the convolution operator and \(z_{j}^{l}\) is the input of the LeakyReLU. At this point, the qubits are folded into the bit state of the computer and passed to the LSTM framework in a one-to-one manner, as the data has already been flattened in the convolutional part. Finally, it is connected to a fully connected layer to fit and predict the transmitted data, and the output is the predicted value at that time. The difference between the predicted and actual values is calculated to obtain the error for evaluating the performance of the model framework. Figure 5 illustrates the framework of the QCNN-LSTM model. The model can accept either discrete data or periodic time-window data as inputs, and different data types will  
Figure 4: Dense angle-coded quantum circuits representing the input data points of the time-window QCNN-LSTM model. be processed differently. The inputs \(X^{1}\) through \(X^{n}\) represent the n feature variables of the dataset. In the context of drift trajectory prediction, these variables include the latitude, longitude, speed, direction, temperature, and other environmental factors of the drifting object. In the input section of the framework, quantum-dense coding transforms data into quantum bits. The entanglement property of quantum bits allows one qubit to transmit multiple classical bits, greatly expanding the channel capacity [39]. Each quantum bit contains two orthogonal states (\(\ket{0}\) and \(\ket{1}\), i.e., a linear combination). If the auxiliary entanglement is in a Bell state, transmitting 1 quantum bit can transmit 2 classical bits of information. Therefore, the entanglement-assisted classical channel capacity is doubled. After the data is input into the framework, spatial features are extracted in the QCNN section, and temporal features are extracted in the LSTM section, giving the model an advantage in spatiotemporal prediction. QCNN is used as the encoder in the encoder-decoder structure. Although QCNN does not directly support sequential input, modifying the convolution layers to be one-dimensional allows it to read sequential input and automatically learn salient features. The LSTM decoder can interpret this data. Matrix multiplication in LSTM is converted into a convolution operation and concatenated with LSTM after one-dimensional convolution.