  Unitary $T$-designs play an important role in quantum information, with
diverse applications in quantum algorithms, benchmarking, tomography, and
communication. Until now, the most efficient construction of unitary
$T$-designs for $n$-qudit systems has been via random local quantum circuits,
which have been shown to converge to approximate $T$-designs in the diamond
norm using $O(T^{5+o(1)} n^2)$ quantum gates. In this work, we provide a new
construction of $T$-designs via random matrix theory using $\tilde{O}(T^2 n^2)$
quantum gates. Our construction leverages two key ideas. First, in the spirit
of central limit theorems, we approximate the Gaussian Unitary Ensemble (GUE)
by an i.i.d. sum of random Hermitian matrices. Second, we show that the product
of just two exponentiated GUE matrices is already approximately Haar random.
Thus, multiplying two exponentiated sums over rather simple random matrices
yields a unitary $T$-design, via Hamiltonian simulation. A central feature of
our proof is a new connection between the polynomial method in quantum query
complexity and the large-dimension ($N$) expansion in random matrix theory. In
particular, we show that the polynomial method provides exponentially improved
bounds on the high moments of certain random matrix ensembles, without
requiring intricate Weingarten calculations. In doing so, we define and solve a
new type of moment problem on the unit circle, asking whether a finite number
of equally weighted points, corresponding to eigenvalues of unitary matrices,
can reproduce a given set of moments.
