To solve the problem of short-term multivariate load forecasting for integrated energy systems, a model based on a quantum weighted multi-hierarchy gated recurrent unit (QWMHGRU) neural network is proposed. The maximum information coefficient is adopted to analyze the correlation between multivariate loads and the relevance between loads and weather factors to form the multiple input sequence. Then the gate structure of the GRU is improved to form a multi-hierarchy gate recurrent unit (MHGRU) and the quantum weighted neurons are introduced into the MHGRU to form the QWMHGRU multivariate load forecasting model. The simulation results show that the weighted average accuracy of the multivariate load forecasting model of QWMHGRU model in summer and winter can reach over 97%. This is higher than MHGRU, QWGRU and GRU models. This work is supported by the National Natural Science Foundation of China (No. 52077108).