Usefulness of the permanent-oracle reduction for classical sampling of boson sampling with general inputs

Investigate the extent to which the strategy of sequentially sampling conditional marginals using an oracle that approximates permanents of complex weighted matrices enables efficient classical sampling for boson sampling with general (not necessarily non-negative) input matrices, despite the NP-hardness of approximating such permanents in general.

Background

The authors’ polynomial-time classical sampling algorithms rely on non-negativity conditions (either on graph-related weights or on matrix entries). For general complex inputs, they outline a reduction that would sample by drawing rows sequentially from their conditional marginals, provided one had access to an oracle capable of approximating the permanents of certain complex weighted matrices.

Because approximating the permanent of complex matrices is NP-hard in general (even for some restricted classes), the practical scope of this reduction-based sampling strategy is unclear. The authors pose an explicit open question regarding the usefulness of this approach, potentially for special cases or restricted matrix families where permanent approximation might be tractable.