Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the ``large $p$, small $n$'' setting, in which the problem dimension $p$ is comparable to or larger than the sample size $n$. This paper studies PCA in this high-dimensional regime, but under the additional assumption that the maximal eigenvector is sparse, say, with at most $k$ nonzero components. We consider a spiked covariance model in which a base matrix is perturbed by adding a $k$-sparse maximal eigenvector, and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector, as follows: (a) a simple diagonal thresholding method, which transitions from success to failure as a function of the rescaled sample size $θ_{\mathrm{dia}}(n,p,k)=n/[k^2\log(p-k)]$; and (b) a more sophisticated semidefinite programming (SDP) relaxation, which succeeds once the rescaled sample size $θ_{\mathrm{sdp}}(n,p,k)=n/[k\log(p-k)]$ is larger than a critical threshold. In addition, we prove that no method, including the best method which has exponential-time complexity, can succeed in recovering the support if the order parameter $θ_{\mathrm{sdp}}(n,p,k)$ is below a threshold. Our results thus highlight an interesting trade-off between computational and statistical efficiency in high-dimensional inference.