aiqbench analysis
This study resolves a long-standing theoretical gap by proving that anticoncentration and relative-error approximate state 2-designs are equivalent properties for local random quantum circuits, overturning the previous assumption of a depth disparity. The finding holds universally across architectures invariant under single-qubit unitaries, simplifying the criteria required to characterize scrambling in these ensembles without needing linear-depth constructions.
Anti-concentration is (almost) all you need
arXiv:2510.23719v3 Announce Type: replace Abstract: Until very recently, it was generally believed that the (approximate) 2-design property is strictly stronger than anticoncentration of random quantum circuits, mainly because it was shown that the latter anticoncentrate 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 anticoncentration. Their result does however not apply to ordinary local random circuits, a gap which we close in this letter, at least for 2-designs. More precisely, we show that anticoncentration 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.
Abstract:Until very recently, it was generally believed that the (approximate) 2-design property is strictly stronger than anticoncentration of random quantum circuits, mainly because it was shown that the latter anticoncentrate 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 anticoncentration. Their result does however not apply to ordinary local random circuits, a gap which we close in this letter, at least for 2-designs. More precisely, we show that anticoncentration 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.
4+2 pages. v3: close to published version, no major changes
arXiv:2510.23719 [quant-ph]
(or arXiv:2510.23719v3 [quant-ph] for this version)
Phys. Rev. Lett. 137, 050601, 2026
https://doi.org/10.1103/z3mp-5gml DOI(s) linking to related resources
Full text under the publisher's open licence · 166 words
