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We propose a mechanism for reaching pseudorandom quantum states, i.e., statesthat computationally indistinguishable from Haar random, with shallow quantumcircuits of depth $\log n$, where $n$ is the number of qudits. While it isoften argued that a $\log n$ ``computational time" provides a lower bound onthe speed of information scrambling, the level of scrambling implied by thosearguments does not rise to the level required for pseudorandomness. Indeed, weshow that $\log n$-depth $2$-qudit-gate-based generic random quantum circuitsthat match the ``speed limit" for scrambling cannot produce computationallypseudorandom quantum states. This conclusion is connected with the presence ofpolynomial (in $n$) tails in the stay probability of short Pauli strings thatsurvive evolution through $\log n$ layers of such circuits. We argue, however,that producing pseudorandom quantum states with shallow $\log n$-depth quantumcircuits can be accomplished if one employs universal families of``inflationary'' quantum (IQ) gates which eliminate the tails in thestay-probability. We prove that IQ-gates cannot be implemented with $2$-qubitgates but can be realized either as a subset of 2-qu$d$it-gates in $U(d^2)$with $d\ge 3$ and $d$ prime, or as special 3-qubit gates. Identifying thefastest way of producing pseudorandom states is conceptually important and hasimplications to many areas of quantum information.

Fast pseudorandom quantum state generators via inflationary quantum gates