Variable-Cliff Nielsen Geometry and an Exponent -4/3 Lower Bound for the Infinite-Cliff Diameter

Let $D=2^n$ and $M=3n+9\left(\genfrac{}{}{0px}{}{n}{2}\right)$. We study the right-invariant one-step-cliff metric $d_Q$ on $\mathrm{PU}(D)$, with unit penalty on Pauli weights one and two and penalty $Q$ on all higher weights. If $M\mathrm{/}{Q}_D\to 0$ and $M{Q}_D^{3\mathrm{/}4}\mathrm{/}{D}^2\to 0$, then, for every fixed $0<x<\pi \mathrm{/}\sqrt{3}$, $${\mu }_D(B_{Q_D}([I],x\sqrt{Q_D}))\le e^{-c_x{D}^2}\mathrm{.}$$ Here ${\mu }_D$ is normalized Haar measure. Thus the Haar-typical distance from the identity and the diameter are both asymptotic to $(\pi \mathrm{/}\sqrt{3})\sqrt{Q_D}$ throughout the window $M\ll Q_D\ll D^{8\mathrm{/}3}{M}^{-4\mathrm{/}3}$. Choosing $Q_D=\kappa D^{8\mathrm{/}3}{M}^{-4\mathrm{/}3}$ with sufficiently small fixed $\kappa >0$ yields a Haar-typical lower bound of order $D^{4\mathrm{/}3}{M}^{-2\mathrm{/}3}$ for the corresponding infinite-cliff Carnot–Carathéodory distance, outside an $e^{-\mathrm{\Omega }(D^2)}$ exceptional set. The infinite-cliff diameter therefore has exponential lower rate at least $4\mathrm{/}3$, disproving Brown’s exponent-one conjecture. The same estimate gives a fixed-error no-ancilla two-qubit circuit lower bound of the same order.