We explore the single-spiked covariance model within the context of sparse principal component analysis (PCA), which aims to recover a sparse unit vector from noisy samples. From an information-theoretic perspective, $O(k \log p)$ observations are sufficient to recover a $k$-sparse $p$-dimensional vector $\mathbf{v}$. However, existing polynomial-time methods require at least $O(k^2)$ samples for successful recovery, highlighting a significant gap in sample efficiency. To bridge this gap, we introduce a novel thresholding-based algorithm that requires only $\Omega(k \log p)$ samples, provided the signal strength $\lambda = \Omega(||\mathbf{v}||_\infty^{-1})$. We also propose a two-stage nonconvex algorithm that further enhances estimation performance. This approach integrates our thresholding algorithm with truncated power iteration, achieving the minimax optimal rate of statistical error under the desired sample complexity. Numerical experiments validate the superior performance of our algorithms in terms of estimation accuracy and computational efficiency.