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

Abstract

Let $D=2^n$ and $M=3n+9\binom n2$. We study the right-invariant one-step-cliff metric $d_Q$ on $\operatorname{PU}(D)$, with unit penalty on Pauli weights one and two and penalty $Q$ on all higher weights. If $M/Q_D\to0$ and $MQ_D^{3/4}/D^2\to0$, then, for every fixed $0<x<π/\sqrt3$, \[ μ_D\bigl(B_{Q_D}([I],x\sqrt{Q_D})\bigr)\le e^{-c_xD^2}. \] Here $μ_D$ is normalized Haar measure. Thus the Haar-typical distance from the identity and the diameter are both asymptotic to $(π/\sqrt3)\sqrt{Q_D}$ throughout the window $M\ll Q_D\ll D^{8/3}M^{-4/3}$. Choosing $Q_D=κD^{8/3}M^{-4/3}$ with sufficiently small fixed $κ>0$ yields a Haar-typical lower bound of order $D^{4/3}M^{-2/3}$ for the corresponding infinite-cliff Carnot--Carathéodory distance, outside an $e^{-Ω(D^2)}$ exceptional set. The infinite-cliff diameter therefore has exponential lower rate at least $4/3$, disproving Brown&#39;s exponent-one conjecture. The same estimate gives a fixed-error no-ancilla two-qubit circuit lower bound of the same order.