Determine the optimal dimension dependence for general-action quantum linear bandits

Determine the optimal dependence on the dimension \(d\) of regret for quantum linear bandits with general, potentially infinite, action sets, between the known \(\Omega(d\log(T/d))\) lower bound and the \(O(d^{3/2}\,\mathrm{polylog}\,T)\) upper bound.

Background

For finite action sets, the paper obtains a nearly linear-in-$d$ regret upper bound when the number of actions is polynomial in $d$. For general action sets, the cited and derived upper bounds are $O(d^2\,\mathrm{polylog}\,T)$ and $O(d^{3/2}\,\mathrm{polylog}\,T)$, while the paper's lower-bound construction gives $\Omega(d\log(T/d))$. The optimal dimension dependence in the general-action setting therefore remains undetermined.