  Uniformly random unitaries, i.e. unitaries drawn from the Haar measure, have
many useful properties, but cannot be implemented efficiently. This has
motivated a long line of research into random unitaries that "look"
sufficiently Haar random while also being efficient to implement. Two different
notions of derandomisation have emerged: $t$-designs are random unitaries that
information-theoretically reproduce the first $t$ moments of the Haar measure,
and pseudorandom unitaries (PRUs) are random unitaries that are computationally
indistinguishable from Haar random.
  In this work, we take a unified approach to constructing $t$-designs and
PRUs. For this, we introduce and analyse the "$PFC$ ensemble", the product of a
random computational basis permutation $P$, a random binary phase operator $F$,
and a random Clifford unitary $C$. We show that this ensemble reproduces
exponentially high moments of the Haar measure. We can then derandomise the
$PFC$ ensemble to show the following:
  (1) Linear-depth $t$-designs. We give the first construction of a
(diamond-error) approximate $t$-design with circuit depth linear in $t$. This
follows from the $PFC$ ensemble by replacing the random phase and permutation
operators with their $2t$-wise independent counterparts.
  (2) Non-adaptive PRUs. We give the first construction of PRUs with
non-adaptive security, i.e. we construct unitaries that are indistinguishable
from Haar random to polynomial-time distinguishers that query the unitary in
parallel on an arbitary state. This follows from the $PFC$ ensemble by
replacing the random phase and permutation operators with their pseudorandom
counterparts.
  (3) Adaptive pseudorandom isometries. We show that if one considers
isometries (rather than unitaries) from $n$ to $n + \omega(\log n)$ qubits, a
small modification of our PRU construction achieves general adaptive security.
