Optimal capacity dependence when the process memory budget grows

Determine the optimal joint dependence of the sequential response capacity on duration, memory budget, and probability margin for quantum process classes in which the propagated memory dimension varies with the horizon.

Background

For fixed propagated memory dimension, the paper proves a tight $K\log((K+1)/\gamma)$ law, while for sufficiently large memory it establishes the saturated exponential-in-duration order. Between these regimes, the paper gives lower and upper bounds, including a quadratic-memory qubit slice with a polynomial gap between them.

The unresolved problem is to close the remaining gap and obtain the sharp capacity law uniformly across growing-memory regimes, jointly accounting for the duration $K$, memory budget $r$, and margin $\gamma$.