This paper proposes a stacked intelligent metasurface (SIM)-assisted holographic MIMO (HMIMO) communication system that optimizes the achievable rate by jointly optimizing the phase shifts of the SIMs at both the transmitter and receiver, as well as the covariance matrix of the transmitted signal. The proposed method uses an iterative projected gradient algorithm to solve a non-convex optimization problem, which guarantees convergence and reduces the number of iterations compared to the alternating optimization (AO) method. Simulation results show that the proposed algorithm achieves the same achievable rate as the AO method but with significantly fewer iterations and lower computational complexity. The system model includes a detailed description of the SIM structure, the channel model, and the optimization problem formulation. The algorithm is analyzed for convergence and computational complexity, and numerical results demonstrate its effectiveness in improving the achievable rate of SIM-assisted HMIMO systems. The results show that the proposed method outperforms conventional MIMO and RIS-assisted systems in terms of achievable rate and convergence speed. The paper concludes that the proposed SIM-assisted HMIMO system is a promising approach for future 6G networks.This paper proposes a stacked intelligent metasurface (SIM)-assisted holographic MIMO (HMIMO) communication system that optimizes the achievable rate by jointly optimizing the phase shifts of the SIMs at both the transmitter and receiver, as well as the covariance matrix of the transmitted signal. The proposed method uses an iterative projected gradient algorithm to solve a non-convex optimization problem, which guarantees convergence and reduces the number of iterations compared to the alternating optimization (AO) method. Simulation results show that the proposed algorithm achieves the same achievable rate as the AO method but with significantly fewer iterations and lower computational complexity. The system model includes a detailed description of the SIM structure, the channel model, and the optimization problem formulation. The algorithm is analyzed for convergence and computational complexity, and numerical results demonstrate its effectiveness in improving the achievable rate of SIM-assisted HMIMO systems. The results show that the proposed method outperforms conventional MIMO and RIS-assisted systems in terms of achievable rate and convergence speed. The paper concludes that the proposed SIM-assisted HMIMO system is a promising approach for future 6G networks.