2025-12-08
Anti-concentration is (almost) all you need
Publication
Publication
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 letter, 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.
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| doi.org/10.48550/arXiv.2510.23719 | |
| Zwaartekracht QSC Ada Lovelace | |
| Organisation | Algorithms and Complexity |
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Heinrich, M., Haferkamp, J., Roth, I., & Helsen, J. (2025). Anti-concentration is (almost) all you need. doi:10.48550/arXiv.2510.23719 |
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