Improve the randomized rounding query complexity

Improve the randomized rounding algorithm of Jia, Laddha, Lee, and Vempala for well-rounding convex bodies from its current \widetilde{O}(d^{3.5}) membership-query complexity to \widetilde{O}(d^3).

Background

The volume-estimation algorithm uses the randomized rounding procedure of Jia, Laddha, Lee, and Vempala as a preprocessing step. The corrected query complexity of that procedure is \widetilde{O}(d{3.5}), which determines the additive term in the quantum volume-estimation bounds discussed in the paper.

Reducing the rounding complexity to \widetilde{O}(d3) would improve the preprocessing cost and potentially sharpen both randomized and quantum volume-estimation algorithms. The paper identifies this reduction as an unresolved problem.

References

To the best of our knowledge, improving the $\widetilde O(d{3.5})$ query complexity of to $\widetilde O(d3)$ remains an open problem.

— Improved quantum volume estimation with transducers and amortized quantum walks  (2609.40217 - Cornelissen et al., 30 Sep 2026) in Footnote following the paragraph on the Cousins–Vempala volume-estimation algorithm, Section 1.3 (Quantum volume estimation)

While this was already considered in (see \cref{footnote:chakrabarti}), it seems largely open. The most immediate technical bottleneck seems to be to develop a non-destructive multivariate mean estimation routine, generalizing the non-destructive univariate quantum mean estimation used in this and prior works. We leave this for future work as well.

— Improved quantum volume estimation with transducers and amortized quantum walks  (2609.40217 - Cornelissen et al., 30 Sep 2026) in Section 1.4, paragraph titled “Quantum rounding”