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The Internet of Things (IoT) has rapidly grown recently, and mobile devices (MDs) have encountered widespread usage. All of these cause an increase in the demand for more powerful computing resources. Meanwhile, a new concept called mobile edge computing (MEC) was introduced as a promising technology to access powerful computing resources closer to the user side for a quick and effective response, especially for time-intensive applications. Task offloading has emerged as a solution to allocate resources among computing resources of smart devices or computational resources available in MEC. This study presents a new binary quantum approach based on an arithmetic optimization algorithm (BQAOA) for computational tasks offloading decisions on MDs with low complexity and guaranteed convergence. However, since task offloading is an NP-hard problem, there is a need to use methods that provide the optimal possible solution for various quality criteria, including response time and energy consumption. Indeed, this is where the advantages of arithmetic optimization algorithms (AOA) and quantum computing have been used to improve the performance of MDs. This paper introduces a 2-tier architecture from the user to the cloud computing server-side. Also, a Markov model is proposed to compute the average network bandwidth in the offloading problem. The proposed BQAOA is compared with the best state-of-the-art algorithms in heuristic and meta-heuristic fields in different scenarios. The simulation results showed 12.5%, 12%, and 26% improvement in energy consumption, makespan, and Energy SLA Violations (ESV) optimization parameters, respectively.