In the vehicle routing problem, the uncertainty of delivery time and customer expectation time greatly affects the selection of the delivery route and the customer service level. Therefore, the traditional vehicle routing optimization model and algorithm with time windows are no longer applicable. The author uses the travel budget time formula to generate a grey time window for vehicle distribution, and uses a probability density function to whiten the Grey Delivery Time, uses a fuzzy gradient function to represent customer satisfaction, and builds a multi-depot with maximum customer satisfaction and minimum cost as the goal. In the process of solving the model, the classical quantum evolutionary algorithm has a problem of a large amount of effective information being lost in the mapping processing between the quantum domain, the binary domain, and the problem domain. The author puts forward the concept of a quantum cell body and operation method of qubit alignment and constructs a new quantum evolution algorithm to solve the model. The author conducts experiments through specific examples to verify the correctness of the model and the effectiveness of the algorithm.