A Method for Validating a Boson Sampler Based on a Radial Function in the Space of Observable States

For the boson sampling problem, a validation method is proposed based on the analysis of the space of measured Fock states expressed as strings of non-negative integers. The method is based on introducing a radial function $${{G}_{{{{p}_{0}}}}}$$ (r) in the space of such strings with an l1-metric. The function $${{G}_{{{{p}_{0}}}}}$$ (r) depends on the internal structure of the collected statistics for strings of integers and will therefore differ significantly for different samplers. The method is tested for samplers with Fock, coherent, and thermal states, as well as for distinguishable particles in the nonunit quantum efficiency regime of the measuring photodetectors. It is shown that for a photon loss probability ploss ≤ 0.1 at the photodetector, the radial function of a sampler with Fock states is qualitatively different from that of samplers with coherent and thermal states. Knowledge of $${{G}_{{{{p}_{0}}}}}$$ (r) in this case is sufficient to qualitatively distinguish a boson sampler from those described above. For ploss ≥ 0.2, the radial functions of the Fock, thermal, and coherent samplers may be superficially similar. Validation in this case is only possible if, in addition to the radial function, the ploss value is known. A statistical test is used for validation. The radial functions $${{G}_{{{{p}_{0}}}}}$$ (r) of a Fock sampler and a sampler with distinguishable particles are qualitatively indistinguishable for any ploss loss level. Therefore, ploss must be known to statistically distinguish these samplers.