Deep learning (DL) is known for its excellence in feature learning and its ability to deliver high-accuracy results. Its application to ECG biometric recognition has received increasing interest but is also accompanied by several deficiencies. In this study, we focus on applying DL, especially convolutional neural networks (CNNs), to ECG biometric identification to address these deficiencies. Using prestored user-specific feature vectors, the proposed scheme can exclude unregistered subjects to realize "open-set" identification. With the help of its scalable structure and "transfer learning," new subjects can be enrolled in an existing system without the need for storing the ECGs of those previously enrolled. Finally, schemes based on the quantum evolutionary algorithm (QEA) are presented to prune unnecessary filters in the proposed CNN model. The performance of the proposed scheme was evaluated using the ECGs of 285 subjects from the PTB dataset. The experimental results demonstrate an identification rate of more than 99% in closed-set identification. Although incorporating the proposed method for unregistered subject exclusion degraded the identification performance slightly, the ability of the approach to resist a dictionary attack was evident. Finally, using the QEA-based filter pruning method and its two-stage extension reduced the number of floating-point operations required to complete one identity recognition to 1.20% and 0.22% of the original value without significantly impacting the identification accuracy.