We study sparse principal components analysis in the high-dimensional setting, where p (the number of variables) can be much larger than n (the number of observations). We prove optimal, non-asymptotic lower and upper bounds on the minimax estimation error for the leading eigenvector when it belongs to an ℓq ball for q∈[0,1]. Our bounds are sharp in p and n for all q∈[0,1] over a wide class of distributions. The upper bound is obtained by analyzing the performance of ℓq-constrained PCA. In particular, our results provide convergence rates for ℓ1-constrained PCA.