Estimation of remaining capacity is essential for ensuring the safety and reliability of lithium-ion batteries. In actual operation, batteries are seldom fully discharged. For a constant current-constant voltage charging mode, the incomplete discharging process affects not only the initial state but also processed variables of the subsequent charging profile, thereby mainly limiting the applications of many feature-based capacity estimation methods which rely on a whole cycling process. Since the charging information of the constant voltage profile can be completely saved whether the battery is fully discharged or not, a geometrical feature of the constant voltage charging profile is extracted to be a new aging feature of lithium-ion batteries under the incomplete discharging situation in this work. By introducing the quantum computing theory into the classical machine learning technique, an integrated quantum particle swarm optimization-based support vector regression estimation framework, as well as its application to characterize the relationship between extracted feature and battery remaining capacity, are presented and illustrated in detail. With the lithium-ion battery data provided by NASA, experiment and comparison results demonstrate the effectiveness, accuracy, and superiority of the proposed battery capacity estimation framework for the not entirely discharged condition.