Accurate and efficient short-term forecasting of multiple loads is of great significance to the operation control and scheduling of integrated energy distribution systems. In order to improve the effect of load forecasting, a mogrifier-quantum weighted memory enhancement long short-term memory (Mogrifier-QWMELSTM) neural network forecasting model is proposed. Compared with the conventional LSTM neural network model, the model proposed in this paper has three improvements in model structure and model composition. First, the mogrifier is added to make the data fully interact with each other. This addition can help enhance the correlation between the front and rear data and improve generalization, which is the main disadvantage of LSTM neural network. Second, the memory enhancement mechanism is added on the forget gate to realize the extraction and recovery of forgotten information. The addition can help improve the gradient transmission ability in the learning process of the neural network, make the neural network remain sensitive to distant data information, and enhance the memory ability. Third, the model is composed of quantum weighted neurons. Compared with conventional neurons, quantum weighted neurons have significant advantages in nonlinear data processing and parallel computing, which help to improve the accuracy of load forecasting. The simulation results show that the weighted mean accuracy of the proposed model can reach more than 97.5% in summer and winter. Moreover, the proposed model has good forecasting effect on seven typical days in winter, which shows that the model has good stability.