Anti-concentration is (almost) all you need

TL;DR

This paper proves that for local random quantum circuits, anti-concentration implies they are also approximate 2-designs, establishing an equivalence that simplifies understanding their properties.

Contribution

It demonstrates that anti-concentration alone suffices to imply approximate 2-designs in local random quantum circuits, closing a gap in previous knowledge.

Findings

Anti-concentration implies approximate 2-designs for local random circuits.

Equivalence holds for any invariant random circuit under local unitaries.

Results apply regardless of circuit architecture.

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.