With the increasing adoption of cloud computing, the deployment and management of business processes over cloud environments have become an essential operation for most enterprises, leading to the emergence of BPaaS (Business Process as a Service) as a new cloud service model. This SaaS-like service, like its ancestors, should be strategically distributed and managed over multiple cloud zones, while taking into account several constraints and conditions (e.g., sensitivity of BPaaS fragments, insecure and untrusted cloud zones, lack of resources, and workload changes). However, current BPaaS approaches are static, which means that they are no longer suitable to manage such enterprise-oriented cloud service model and to deal with the uncertain and dynamic nature of cloud availability zones. To fill this gap, we adopt a predictive BPaaS management strategy by proposing a model that forecasts the next-short time overload of cloud zones. These latter, as hosting environments for the managed BPaaS, are categorized as overloaded or underloaded, which triggers the migration of BPaaS fragments to high-performance cloud zones. The proposed neural network prediction model (called QGA-NN) is enhanced with a quantum genetic algorithm to optimize the prediction of cloud zones' overload. QGA-NN is evaluated using a BPaaS placement algorithm, which we defined as a triggered management operation. Experimental results have proved the accuracy and effectiveness of our predictive approach, compared with state-of-the-art solutions.