We prove a Carbery-Wright style anti-concentration inequality for the unitary
Haar measure, by showing that the probability of a polynomial in the entries of
a random unitary falling into an $\varepsilon$ range is at most a polynomial in
$\varepsilon$. Using it, we show that the scrambling speed of a random quantum
circuit is lower bounded: Namely, every input qubit has an influence that is at
least inverse exponential in depth, on any output qubit touched by its
lightcone. Our result on scrambling speed works with high probability over the
choice of a circuit from an ensemble, as opposed to just working in
expectation.

As an application, we give the first polynomial-time algorithm for learning
log-depth random quantum circuits with Haar random gates up to polynomially
small diamond distance, given oracle access to the circuit. Other applications
of this new scrambling speed lower bound include:

$\bullet$ An optimal $\Omega(\log \varepsilon^{-1})$ depth lower bound for
$\varepsilon$-approximate unitary designs on any circuit architecture;

$\bullet$ A polynomial-time quantum algorithm that computes the depth of a
bounded-depth circuit, given oracle access to the circuit.

Our learning and depth-testing algorithms apply to architectures defined over
any geometric dimension, and can be generalized to a wide class of
architectures with good lightcone properties.

Updated technical overview, fixed a mistake in the last section

We prove a Carbery-Wright style anti-concentration inequality for the unitary Haar measure, by showing that the probability of a polynomial in the entries of a random unitary falling into an $\varepsilon$ range is at most a polynomial in $\varepsilon$. Using it, we show that the scrambling speed of a random quantum circuit is lower bounded: Namely, every input qubit has an influence that is at least exponentially small in depth, on any output qubit touched by its lightcone.

We give three applications of this new scrambling speed lower bound that apply to random quantum circuits with Haar random gates:
$\bullet$ An optimal $\Omega(\log \varepsilon^{-1})$ depth lower bound for $\varepsilon$-approximate unitary designs;
$\bullet$ A polynomial-time quantum algorithm that computes the depth of a bounded-depth circuit, given oracle access to the circuit;
$\bullet$ A polynomial-time algorithm that learns log-depth circuits up to polynomially small diamond distance, given oracle access to the circuit.
The first depth lower bound works against any architecture. The latter two algorithms apply to architectures defined over any geometric dimension, and can be generalized to a wide class of architectures with good lightcone properties.