Quantum supremacy: shallow depth Haar random circuits and unitary designs

I had a confusion about shallow depth Haar random quantum circuits. In this paper, in Section B (related works), it is mentioned that Haar random quantum circuits form approximate $$2$$-designs only after $$\mathcal{O}\left(n^{1/D}\right)$$ depth, where $$D$$ is the spatial dimension (proved here).

What does this mean for the output distribution of Haar random quantum circuits that have constant and logarithmic depth? For every fixed string $$z \in \{0, 1\}^{n}$$, does every output probability $$|\langle z| U |0^{n} \rangle|^{2}$$ approximately follow a Porter-Thomas distribution, even for a shallow depth Haar-random $$U$$?

If so, what extra mileage does the $$2$$-design property give us?

• Strictly speaking a "Haar-random unitary circuit" consists of a single multi-qubit Haar-random unitary. Nobody can do this. "Random quantum circuits" (Dalzell et al.) are composed of other random quantum gates. For instance, we can take 2-local Haar-random unitaries of a certain depth. This converges quickly to the multi-qubit Haar measure with depth, however, we are often only interested in the moments. This is strictly weaker and formalised by approximate designs. Harrow-Mehraban showed that geometrically local circuits form approximate t-designs in depth $\mathrm{poly}(t)n^{1/D}$. Commented Feb 22, 2021 at 10:01
• @BlackHat18 As I wrote above, random quantum circuits generally converge to $t$-designs with increasing depth. So they first become 1-designs, then 2-designs and so on. But you definitely need a non-constant depth to be any design, the precise scaling depends on your notion of approximation. Commented Oct 18, 2021 at 8:59