  A central roadblock to analyzing quantum algorithms on quantum states is the
lack of a comparable input model for classical algorithms. Inspired by recent
work of the author [E. Tang, STOC'19], we introduce such a model, where we
assume we can efficiently perform $\ell^2$-norm samples of input data, a
natural analogue to quantum algorithms that assume efficient state preparation
of classical data. Though this model produces less practical algorithms than
the (stronger) standard model of classical computation, it captures versions of
many of the features and nuances of quantum linear algebra algorithms. With
this model, we describe classical analogues to Lloyd, Mohseni, and Rebentrost's
quantum algorithms for principal component analysis [Nat. Phys. 10, 631 (2014)]
and nearest-centroid clustering [arXiv:1307.0411]. Since they are only
polynomially slower, these algorithms suggest that the exponential speedups of
their quantum counterparts are simply an artifact of state preparation
assumptions.
