Extension of GQSP to arbitrary-matrix singular-value transformations

Determine whether generalized quantum signal processing can be efficiently extended to the singular-value domain for arbitrary matrices, in analogy with the extension of quantum signal processing to quantum singular value transformation, while removing the parity and real-coefficient constraints.

Background

The paper identifies the extension of generalized quantum signal processing (GQSP) from unitary or structured signals to arbitrary matrices and their singular values as its most pressing open problem. Standard QSP was generalized to QSVT to support singular-value transformations, but GQSP currently operates under different structural constraints, including a unitary-signal formulation and broader polynomial freedom. An efficient generalized quantum singular value transform that preserves GQSP’s ability to handle non-definite-parity and complex-coefficient polynomials would provide a more flexible unified signal-processing framework.

References

The most pressing open problem remains whether GQSP can be extended into the singular value domain, with arbitrary matrices, in the same way as QSP was extended into QSVT. Since the two frameworks work under fundamentally different constraints, a generalized quantum singular value transform which removes the parity and real-coefficient constraints would be the gold standard signal processing approach. However, it is not known whether an efficient extension which resolves these constraints is possible.

— 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”