Joint time–precision dependence of geometrically local Hamiltonian simulation depth

Determine the optimal joint dependence on evolution time and simulation precision of the circuit depth required for geometrically local Hamiltonian simulation, resolving the gap between the additive time-and-precision lower bound and the multiplicative logarithmic overhead in the upper bound.

Background

The paper establishes an upper bound of order $\mathcal T\log(2+n\mathcal T/\epsilon)$ and a lower bound containing additive time and precision contributions, as summarized in Eq. (joint-depth-comparison). At fixed time, the bounds differ only by a doubly logarithmic factor, but for growing evolution time the comparison leaves a more substantial structural gap: the lower bound adds the time and precision scales, whereas the upper bound multiplies the evolution time by a logarithmic overhead. The authors explicitly identify the optimal combined dependence as unresolved.