Efficient complementary-polynomial finding

Improve the efficiency of algorithms for finding complementary polynomials required for QSP and QSVT phase synthesis, including approaches based on numerical optimization, root finding, factorization, and fast Fourier transforms.

Background

QSP and QSVT phase synthesis requires constructing a complementary polynomial compatible with the target polynomial and the unitary signal-processing constraints. The paper explains that recent advances have reduced the cost of phase-angle synthesis, shifting a major computational bottleneck toward complementary-polynomial construction. Existing approaches include numerical optimization, root finding, factorization, and FFT-based methods, but their efficiency remains an unresolved research issue.

References

It remains an open area of research to improve the efficiency of complementary polynomial finding algorithms.

— From Block-encoding to Generalized Quantum Signal Processing: Principles, Algorithms and Applications  (2609.09977 - Gurfinkel et al., 9 Sep 2026) in Section 6, “Open Problems, Future Directions and Conclusion,” subsection “Open Problems and Future Directions”