Computational complexity of exterior products and multiparticle amplitudes of noninteracting fermions in entangled states
AbstractNoninteracting bosons were proposed to be used for a demonstration of quantum-computing supremacy in a boson-sampling setup. A similar demonstration with fermions would require that the fermions are initially prepared in an entangled state. I suggest that pairwise entanglement of fermions would be sufficient for this purpose. Namely, it is shown that computing multiparticle scattering amplitudes for fermions entangled pairwise in groups of four single-particle states is #P-hard. In linear algebra, such amplitudes are expressed as exterior products of two-forms of rank 2. In particular, a permanent of a N×N matrix may be expressed as an exterior product of N2 two forms of rank 2 in dimension 2N2, which establishes the #P-hardness of the latter.