Fourier-rank lower bounds for further continuous query problems

Develop thresholded and approximate Fourier-rank measures, analogous to threshold degree and approximate degree, and apply them to additional continuous quantum query problems beyond determinant computation, eigenvalue estimation, and convex optimization.

Background

The paper introduces Fourier rank as a continuous analogue of polynomial degree for analyzing quantum query algorithms whose hidden input is a real matrix. The method proves that low-query algorithms have output probabilities supported on low-rank Fourier frequencies, which are orthogonal to the determinant.

The authors note that this framework suggests stronger notions corresponding to threshold degree and approximate degree. Developing such measures and using them to derive lower bounds for other continuous query models is identified as future work rather than established in the paper.

References

We leave the development of these measures and their application to other continuous query problems for future work.

— Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank  (2609.09035 - Augustino et al., 8 Sep 2026) in Paragraph preceding Section 1, Introduction, paragraph titled “The Fourier-rank polynomial method”