  Efficiently sampling a quantum state that is hard to distinguish from a truly
random quantum state is an elementary task in quantum information theory that
has both computational and physical uses. This is often referred to as
pseudorandom (quantum) state generator, or PRS generator for short.
  In existing constructions of PRS generators, security scales with the number
of qubits in the states, i.e.\ the (statistical) security parameter for an
$n$-qubit PRS is roughly $n$. Perhaps counter-intuitively, $n$-qubit PRS are
not known to imply $k$-qubit PRS even for $k<n$. Therefore the question of
\emph{scalability} for PRS was thus far open: is it possible to construct
$n$-qubit PRS generators with security parameter $\lambda$ for all $n,
\lambda$. Indeed, we believe that PRS with tiny (even constant) $n$ and large
$\lambda$ can be quite useful.
  We resolve the problem in this work, showing that any quantum-secure one-way
function implies scalable PRS. We follow the paradigm of first showing a
\emph{statistically} secure construction when given oracle access to a random
function, and then replacing the random function with a quantum-secure
(classical) pseudorandom function to achieve computational security. However,
our methods deviate significantly from prior works since scalable pseudorandom
states require randomizing the amplitudes of the quantum state, and not just
the phase as in all prior works. We show how to achieve this using Gaussian
sampling.
