Most random circuit geometries form approximate 2-designs in O(log n) depth with explicit constants; bridge/lollipop graphs need Ω(n²) gates, and 10-20 layers suffice for 50-qubit near-random circuits.
Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions
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Apparent Universal Behavior in Second Moments of Random Quantum Circuits
Most random circuit geometries form approximate 2-designs in O(log n) depth with explicit constants; bridge/lollipop graphs need Ω(n²) gates, and 10-20 layers suffice for 50-qubit near-random circuits.