Reducing ancilla overhead for the full rational class

Determine whether the full class of rational functions with poles at prescribed conjugate pairs, each of order at most d, can be realized with fewer ancillas while keeping the number of queries proportional to the degree.

Background

Products of Cayley factors provide unitary rational encodings whose numerators are constrained by their denominators. The full rational class with poles at prescribed conjugate pairs can instead be obtained by linearly combining m generalized quantum signal-processing circuits, but that construction requires additional ancillas, specifically ceil(log_2 m) further ancillas.

The paper explicitly leaves unresolved whether comparable expressive power can be achieved with fewer ancillas without sacrificing degree-proportional query complexity. Resolving this would narrow the gap between single-unitary Cayley processing and general linear combinations of resolvents.

References

Whether the full class can be reached with fewer ancillas, keeping the number of queries proportional to the degree, is a question we leave to future work.

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