Fractional evolution with uncontrolled oracle access

Determine whether the matching upper bound for implementing U^t with spectral gap delta remains valid when algorithms have access only to U and U^dagger, without controlled-U access.

Background

The query model analyzed in the paper permits controlled-U and controlled-U^dagger queries, and the paper’s lower bound also applies to the more restrictive model using only U and U^dagger. However, controlled access is used by the known matching upper-bound constructions.

The authors note that controlled-U cannot generally be synthesized from U alone, while existing control-free constructions require an eigenvector ancilla or provide only average-case guarantees. It therefore remains unresolved whether the optimal worst-case scaling established in the controlled-access model can be achieved without controlled queries.