Aperture-level simultaneous transmit and receive (ALSTAR) attempts to utilize adaptive digital transmit and receive beamforming and digital self-interference cancellation methods to establish isolation between the transmit and receive apertures of the single-phase array. However, the existing methods only discuss the isolation of ALSTAR and ignore the radiation efficiency of the transmitter and the sensitivity of the receiver. The ALSTAR array design lacks perfect theoretical support and simplified engineering implementation. This paper proposes an adaptive random group quantum brainstorming optimization (ARGQBSO) algorithm to simplify the array design and improve the overall performance. ARGQBSO is derived from BSO and has been ameliorated in four aspects of the ALSTAR array, including random grouping, initial value presets, dynamic probability functions, and quantum computing. The transmit and receive beamforming carried out by ARGQBSO is robust to all elevation angles, which reduces complexity and is conducive to engineering applications. The simulated results indicate that the ARGQBSO algorithm has an excellent performance, and achieves 166.8 dB of peak EII, 47.1 dBW of peak EIRP, and −94.6 dBm of peak EIS with 1000 W of transmit power in the scenario of an 8-element array.