  In this work we give an efficient construction of unitary $k$-designs using
$\tilde{O}(k\cdot poly(n))$ quantum gates, as well as an efficient construction
of a parallel-secure pseudorandom unitary (PRU). Both results are obtained by
giving an efficient quantum algorithm that lifts random permutations over
$S(N)$ to random unitaries over $U(N)$ for $N=2^n$. In particular, we show that
products of exponentiated sums of $S(N)$ permutations with random phases
approximately match the first $2^{\Omega(n)}$ moments of the Haar measure. By
substituting either $\tilde{O}(k)$-wise independent permutations, or
quantum-secure pseudorandom permutations (PRPs) in place of the random
permutations, we obtain the above results. The heart of our proof is a
conceptual connection between the large dimension (large-$N$) expansion in
random matrix theory and the polynomial method, which allows us to prove query
lower bounds at finite-$N$ by interpolating from the much simpler large-$N$
limit. The key technical step is to exhibit an orthonormal basis for
irreducible representations of the partition algebra that has a low-degree
large-$N$ expansion. This allows us to show that the distinguishing probability
is a low-degree rational polynomial of the dimension $N$.
