Combine the favorable action-set and dimension dependencies of the two elimination algorithms

Improve the regret guarantee for finite-action quantum linear bandits with superpolynomially many actions by combining the nearly linear dimension dependence of LV-G-Elim with the reduced action-count dependence of QMC-G-Elim, potentially achieving a dependence on the number of actions closer to the classical \(\sqrt{\log K}\) factor without incurring the \(d^{3/2}\) dimension factor.

Background

For finite action sets, LV-G-Elim achieves nearly linear dependence on dd but incurs an additional log⁡K\log K factor. The QMC-G-Elim variant removes this log⁡K\log K dependence but has a d3/2d^{3/2} dimension factor. The paper explicitly asks whether these two advantages can be obtained simultaneously, especially when the action set is superpolynomial in the dimension.

References

For finite action sets with K superpolynomial in d, the nearly linear bound of LV-G-Elim carries an extra \log K factor, while the QMC variant removes this dependence at the cost of a d{3/2} dimension factor. Whether the two advantages can be combined, for instance by improving the \log K dependence toward the classical \sqrt{\log K}, remains open.

— Quantum Multi-Armed Bandits and Linear Bandits: Lower Bounds and Algorithms  (2608.14319 - Liu et al., 14 Aug 2026) in Section 7, Conclusion