The key features of 5G, particularly URLLC and mMTC, are designed to achieve low latency and high scalability. To support these requirements, a proposed approach for random access scenarios, known as the GF scheme, eliminates handshakes between the BS and mobile users to reduce latency and accommodate a large number of devices. However, this introduces a new challenge known as AUD, where users send messages indicating their activity status, requiring the BS to detect and decode them. Decoding these messages, however, demands high computational complexity, which increases exponentially with the number of users. Several algorithms, such as ZF and CCR, have been proposed to mitigate this complexity, but they suffer from suboptimal performance. On the other hand, the optimal solution, known as ML, performs well but suffers from high complexity. To address this, quantum algorithms, like Grover's algorithm, have been proposed due to their ability to reduce search complexity while also keep detecting active users performance better. However, Grover's algorithm requires adaptation in this context, as the optimal number of iterations depends on the number of solutions, which is always unknown in the AUD case. To address this, the BBHT and DHA algorithms have been proposed to minimize complexity when the solution is unknown, but both still exhibit relatively high computational demands. In this paper, we propose an Enhanced BBHT that aims to reduce this complexity while maintaining detection performance. Our findings demonstrate that the Enhanced BBHT reduces computational complexity while keeping the performance stable.