Anti-concentration is (almost) all you need
Markus Heinrich, Jonas Haferkamp, Ingo Roth, Jonas Helsen
TL;DR
The paper establishes that anti-concentration of local random quantum circuits implies they form relative-error approximate state-2-designs, achieving this in logarithmic depth for LU-invariant ensembles. It derives an explicit bound $Z_\nu \le (1+\varepsilon) Z_\mathrm{H}$ leading to $\varepsilon' = 2 \frac{q^n+1}{q^n-q} \frac{\varepsilon}{1-q^{-1}}$, so that $|\mathrm{tr}(A \mathsf{m}_\nu) - \mathrm{tr}(A \mathsf{m}_\mathrm{H})| \le \varepsilon' \mathrm{tr}(A \mathsf{m}_\mathrm{H})$ for any psd $A$. This result shows that local random circuits converge to state-2-designs as fast as coarse-grained counterparts, highlighting anti-concentration as a universal proxy for second-moment randomness in many architectures. The authors discuss limitations for unitary designs and note that anti-concentration does not universally guarantee unitary-2-design properties, pointing to subtle distinctions between state and unitary designs. Overall, the work provides a simple, powerful link between collision probability and state-design convergence, with practical implications for analyzing brickwork and other LU-invariant random circuits.
Abstract
Until very recently, it was generally believed that the (approximate) 2-design property is strictly stronger than anti-concentration of random quantum circuits, mainly because it was shown that the latter anti-concentrate in logarithmic depth, while the former generally need linear depth circuits. This belief was disproven by recent results which show that so-called relative-error approximate unitary designs can in fact be generated in logarithmic depth, implying anti-concentration. Their result does however not apply to ordinary local random circuits, a gap which we close in this paper, at least for 2-designs. More precisely, we show that anti-concentration of local random quantum circuits already implies that they form relative-error approximate state 2-designs, making them equivalent properties for these ensembles. Our result holds more generally for any random circuit which is invariant under local (single-qubit) unitaries, independent of the architecture.