  We study the problem of learning mixtures of linear classifiers under
Gaussian covariates. Given sample access to a mixture of $r$ distributions on
$\mathbb{R}^n$ of the form $(\mathbf{x},y_{\ell})$, $\ell\in [r]$, where
$\mathbf{x}\sim\mathcal{N}(0,\mathbf{I}_n)$ and
$y_\ell=\mathrm{sign}(\langle\mathbf{v}_\ell,\mathbf{x}\rangle)$ for an unknown
unit vector $\mathbf{v}_\ell$, the goal is to learn the underlying distribution
in total variation distance. Our main result is a Statistical Query (SQ) lower
bound suggesting that known algorithms for this problem are essentially best
possible, even for the special case of uniform mixtures. In particular, we show
that the complexity of any SQ algorithm for the problem is
$n^{\mathrm{poly}(1/\Delta) \log(r)}$, where $\Delta$ is a lower bound on the
pairwise $\ell_2$-separation between the $\mathbf{v}_\ell$'s. The key technical
ingredient underlying our result is a new construction of spherical designs
that may be of independent interest.
