Determine the gate-count overhead of scheduled trotterization

Determine whether scheduled trotterization for integer-valued gate coefficients requires more elementary gates than the corresponding fixed-weight trotterization, and characterize the behavior of the gate-count difference across target dimension and pulse scale.

Background

For gate synthesis, admissible coefficients are integer-valued because they represent repeated applications of elementary gates and their inverses. Scheduled trotterization adjusts these integer coefficients from step to step to reduce coherent error accumulation, whereas fixed-weight trotterization repeats the same rounded coefficient vector.

The paper defines the gate-count difference between scheduled and fixed-weight implementations and reports numerical evidence for random SU(2) targets showing an average difference close to zero, although the standard deviation appears to grow with the number of steps. The authors do not establish whether this behavior persists generally or how it depends on the Hilbert-space dimension and the scale of the available Hamiltonians.