Variable-Cliff Nielsen Geometry and an Exponent -4/3 Lower Bound for the Infinite-Cliff Diameter
Abstract
Let D=2n and M=3n+9binom n2. We study the right-invariant one-step-cliff metric dQ on PU(D), with unit penalty on Pauli weights one and two and penalty Q on all higher weights. If M/QD→0 and MQD3/4/D2→0, then, for every fixed 0<x<π/√3, μD(BQD([I],x√(QD)))≤ e−cxD2. Here μD is normalized Haar measure. Thus the Haar-typical distance from the identity and the diameter are both asymptotic to (π/√3)√(QD) throughout the window M≪ QD≪ D8/3M−4/3. Choosing QD=κD8/3M−4/3 with sufficiently small fixed κ>0 yields a Haar-typical lower bound of order D4/3M−2/3 for the corresponding infinite-cliff Carnot--Carathéodory distance, outside an e−Ω(D2) exceptional set. The infinite-cliff diameter therefore has exponential lower rate at least 4/3, disproving Brown's exponent-one conjecture. The same estimate gives a fixed-error no-ancilla two-qubit circuit lower bound of the same order.
Related
- Regularity and uniqueness to multi-phase problem with variable exponent
- The space of equators has infinite Hofer's diameter
- Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry
- Fractional gradients and nonlocal PDEs in variable exponent spaces
- Calderón's complex interpolations of variable-exponent Fofana spaces
BibTeX
@misc{fang2026variablecliff,
title = {{Variable-Cliff Nielsen Geometry and an Exponent -4/3 Lower Bound for the Infinite-Cliff Diameter}},
author = {Honghuai Fang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03323},
eprint = {2609.03323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03323},
}
RIS
TY - UNPB
TI - Variable-Cliff Nielsen Geometry and an Exponent -4/3 Lower Bound for the Infinite-Cliff Diameter
AU - Honghuai Fang
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.03323
UR - https://arxiv.org/abs/2609.03323
AB - Let D=2^n and M=3n+9binom n2. We study the right-invariant one-step-cliff metric d_Q on PU(D), with unit penalty on Pauli weights one and two and penalty Q on all higher weights. If M/Q_D→0 and MQ_D^3/4/D^2→0, then, for every fixed 0<x<π/√3, μ_D(B_Q_D([I],x√(Q_D)))≤ 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/√3)√(Q_D) throughout the window M≪ Q_D≪ D^8/3M^−4/3. Choosing Q_D=κD^8/3M^−4/3 with sufficiently small fixed κ>0 yields a Haar-typical lower bound of order D^4/3M^−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's exponent-one conjecture. The same estimate gives a fixed-error no-ancilla two-qubit circuit lower bound of the same order.
ER -