  We prove a \emph{query complexity} lower bound on rank-one principal
component analysis (PCA). We consider an oracle model where, given a symmetric
matrix $M \in \mathbb{R}^{d \times d}$, an algorithm is allowed to make $T$
\emph{exact} queries of the form $w^{(i)} = Mv^{(i)}$ for $i \in
\{1,\dots,T\}$, where $v^{(i)}$ is drawn from a distribution which depends
arbitrarily on the past queries and measurements $\{v^{(j)},w^{(j)}\}_{1 \le j
\le i-1}$. We show that for a small constant $\epsilon$, any adaptive,
randomized algorithm which can find a unit vector $\widehat{v}$ for which
$\widehat{v}^{\top}M\widehat{v} \ge (1-\epsilon)\|M\|$, with even small
probability, must make $T = \Omega(\log d)$ queries. In addition to settling a
widely-held folk conjecture, this bound demonstrates a fundamental gap between
convex optimization and "strict-saddle" non-convex optimization of which PCA is
a canonical example: in the former, first-order methods can have dimension-free
iteration complexity, whereas in PCA, the iteration complexity of
gradient-based methods must necessarily grow with the dimension. Our argument
proceeds via a reduction to estimating the rank-one spike in a deformed Wigner
model. We establish lower bounds for this model by developing a "truncated"
analogue of the $\chi^2$ Bayes-risk lower bound of Chen et al.
