Gate-level comparison of structured Cayley realizations

Derive concrete gate-count comparisons between structured realizations of the Cayley encoding, QSVT applied to a block encoding of the matrix, and quantum eigenvalue transformation of a Trotterized matrix exponential, including the unresolved balance between directly processing a block-encoded resolvent and first converting it into a unitary encoding.

Background

The paper’s main complexity results are stated primarily in query complexity, while only selected structured matrix families receive an elementary-gate analysis. A complete gate-level comparison would quantify whether the query advantage of the Cayley model survives circuit-realization costs across relevant structured families.

The paper also notes an alternative in which a block-encoded, normal shifted resolvent is processed directly through quantum eigenvalue processing, avoiding the amplification needed to make the Cayley encoding unitary. The relative performance of these alternatives is not resolved.

References

On structured families one may also process the block-encoded resolvent directly, which is normal but not Hermitian, through quantum eigenvalue processing: this avoids the amplification that makes the encoding unitary, at the price of a polynomial acting on a smaller spectral region, and the balance between the two remains to be assessed.

— Cayley access models for matrix functions: reachable class and cost  (2610.23429 - Corso, 16 Oct 2026) in Section 6, “Conclusions and open problems”