  In this paper, we aim to address the challenge of hybrid mobile edge-quantum
computing (MEQC) for sustainable task offloading scheduling in mobile networks.
We develop cost-effective designs for both task offloading mode selection and
resource allocation, subject to the individual link latency constraint
guarantees for mobile devices, while satisfying the required success ratio for
their computation tasks. Specifically, this is a time-coupled offloading
scheduling optimization problem in need of a computationally affordable and
effective solution. To this end, we propose a deep reinforcement learning
(DRL)-based Lyapunov approach. More precisely, we reformulate the original
time-coupled challenge into a mixed-integer optimization problem by introducing
a penalty part in terms of virtual queues constructed by time-coupled
constraints to the objective function. Subsequently, a Deep Q-Network (DQN) is
adopted for task offloading mode selection. In addition, we design the Deep
Deterministic Policy Gradient (DDPG)-based algorithm for partial-task
offloading decision-making. Finally, tested in a realistic network setting,
extensive experiment results demonstrate that our proposed approach is
significantly more cost-effective and sustainable compared to existing methods.
