Extend the design-based quantum linear-bandit analysis to bounded-variance rewards

Develop a complete deterministic-budget estimator and use it to extend the LV-G-Elim design-based regret analysis from bounded rewards to the bounded-variance quantum reward-oracle setting, with the confidence radii and query allocations appropriately scaled by the known variance bound.

Background

The upper-bound analysis in the paper assumes rewards are bounded in $[0,1]$, primarily because the quantum mean-estimation primitive is stated under that condition. The authors explain that existing bounded-variance estimators suggest an extension with bias and variance scaling with a known variance bound, but the available procedure does not yet provide the deterministic query-budget control required by their design-based elimination algorithm. A complete bounded-variance treatment is therefore left unresolved.