  We propose a mechanism for reaching pseudorandom quantum states,
computationally indistinguishable from Haar random, with shallow log-n depth
quantum circuits, where n is the number of qudits. We argue that $\log n$ depth
2-qubit-gate-based generic random quantum circuits that are claimed to provide
a lower bound on the speed of information scrambling, cannot produce
computationally pseudorandom quantum states. This conclusion is connected with
the presence of polynomial (in $n$) tails in the stay probability of short
Pauli strings that survive evolution through such shallow circuits. We show,
however, that stay-probability-tails can be eliminated and pseudorandom quantum
states can be accomplished with shallow $\log n$ depth circuits built from a
special universal family of `inflationary' quantum (IQ) gates. We prove that
IQ-gates cannot be implemented with 2-qubit gates, but can be realized either
as a subset of 2-qudit-gates in $U(d^2)$ with $d\ge 3$ and $d$ prime, or as
special 3-qubit gates.
