Extend block encoding beyond generator-representable structure

Establish an extension of the block-encoding construction for generator-representable semiseparable matrices to semiseparable matrix structures that are not generator-representable.

Background

The paper constructs a direct block encoding for the smallest class in the stated hierarchy: generator-representable, or one-pair, semiseparable matrices. The conclusion notes that extensions to r-semiseparable matrices and generator-representable-plus-diagonal matrices appear straightforward using combinations of the presented construction.

By contrast, the authors explicitly identify the extension to non-generator-representable structure as unresolved. This concerns semiseparable or related rank-structured matrices whose off-diagonal structure cannot be described by the two global generator vectors used by the paper's circuit.

References

However, the extension to non-generator-representable structure is still an open problem.

— Block encoding of semiseparable matrices via their generators  (2610.13228 - Antonioli et al., 9 Oct 2026) in Section "Conclusions"

We do not know examples of semiseparable-plus-diagonal matrices of interest in applications whose Cayley transform is a CMV matrix with coefficients of this kind, so this construction is presented as a possible direction rather than as a realized one.

— Cayley access models for matrix functions: reachable class and cost  (2610.23429 - Corso, 16 Oct 2026) in Section 5.3, “The reverse direction: structured unitaries”; reiterated in Section 6, “Conclusions and open problems”