Dependence of query complexity on the fractional power

Prove that the fractional query complexity satisfies Q=Theta((1/delta) log(sin(pi t)/epsilon)) uniformly for epsilon <= c sin(pi t), including an upper-bound construction whose cost adapts to the size of the discontinuity jump.

Background

The paper establishes the optimal query complexity Theta_tau((1/delta) log(1/epsilon)) only when the fractional power parameter t is bounded away from 0 and 1 by a fixed constant tau. The authors observe that their lower bound becomes ineffective when the approximation error is comparable to sin(pi t), consistently with the fact that the identity or a single query already approximates Ut up to global phase when t is close to an endpoint.

The unresolved problem is to determine the finer dependence on t and sin(pi t), and specifically to construct an upper bound whose query cost reflects the smaller jump across the branch cut when t approaches 0 or 1.

References

We conjecture that $Q=\Theta!\left(\frac{1}{\delta}\log\frac{\sin(\pi t)}{\varepsilon}\right)$ uniformly for $\varepsilon\le c\sin(\pi t)$. Proving this requires an upper bound whose cost adapts to the size of the jump.

— Optimal query complexity for fractional quantum evolution  (2610.01940 - Liu et al., 1 Oct 2026) in Section Discussion and outlook, paragraph “Dependence on t”

A natural conjecture is a cost of $\Theta!\left(\frac{1}{\delta}\log\frac{J}{\varepsilon}\right)$ whenever $f$ is analytic on the arc, where $J$ is the jump of $e{if}$ across the gap. Establishing this would give a unified lower bound theory for phase functions of unitary oracles.

— Optimal query complexity for fractional quantum evolution  (2610.01940 - Liu et al., 1 Oct 2026) in Section Discussion and outlook, paragraph “Other spectral transformations”