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In wireless sensor networks, a large amount of existing routing protocols consider energy efficiency as the main objective and assume data traffic with unconstrained delivery requirements. However, the introduction of image and video sensors demands certain quality of service (QoS) from the routing protocols and underlying networks. Finding routing paths that satisfy different metrics of the QoS requirement is a NP-complete problem. In this paper, a quantum genetic algorithm based energy-aware QoS routing protocol (QG-QoS) is proposed for wireless sensor networks, which can run efficiently with best-effort traffic. To the best of our knowledge, QG-QoS is the first quantum genetic algorithm based QoS routing protocol in wireless sensor networks. Simulation results show that our approach is better than previous protocols for different metrics.