We construct ε-approximate unitary k-designs on n qubits in circuit depth O(logkloglognk/ε). The depth is exponentially improved over all known results in all three parameters
Nearby in the stack
n
,
k
,
ε
. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses
O~(nk)
ancilla qubits and
O(nk)
bits of randomness, which are also optimal up to
log(nk)
factors. An alternative construction achieves a smaller ancilla count
O~(n)
with circuit depth
O(kloglognk/ε)
. To achieve these efficient unitary designs, we introduce a highly-structured random unitary ensemble that leverages long-range two-qubit gates and low-depth implementations of random classical hash functions. We also develop a new analytical framework for bounding errors in quantum experiments involving many queries to random unitaries. As an illustration of this framework's versatility, we provide a succinct alternative proof of the existence of pseudorandom unitaries.