Heisenberg scaling with constant time resolution

Determine whether a Hamiltonian learning algorithm can achieve Heisenberg-limited total evolution time O(1/ε) while maintaining constant time resolution (i.e., the minimum controllable time step independent of ε).

Background

Two key resource metrics are total evolution time (query complexity) and time resolution (smallest time increment required for control). Previous algorithms achieve either O(1/ε) with time resolution Θ(√ε) or O(1/ε^2) with constant time resolution.

It has been conjectured that achieving both Heisenberg scaling and constant time resolution is possible, motivating a unified approach that avoids the trade-off.