Complexity classification of balanced slice design
Determine whether the balanced slice-design optimization problem is weakly or strongly NP-hard beyond the established weak NP-completeness of its restricted two-slice positive-integer decision version.
References
The reduction establishes NP-hardness of the unrestricted minimization problem but leaves its classification as weakly or strongly NP-hard open.
We do not address the effect of circuit slice ordering on communication cost or Trotter error. Necaise et al. do suggest a pathfinding approach to order slices such that communication cost is minimised. One further consideration which is left open in our method is the effect of overlap in the sets of parallelizable terms of encodings used. Our method removes parallelized Hamiltonian terms with each iteration, leaving open the possibility that some terms would be parallelized by more than one encoding. This is certainly the case, and clearly worth considering, if using a covering design which covers terms more than once, as (16n,8n,4) coverings do. Assuming all parallelized slices are implemented without error, Trotter error resulting from partitioning the Hamiltonian into slices still remains. This may be minimised both by ordering slices, and selecting which slice terms are assigned to. We leave this open for future work.