Computational complexity of exterior products and multi-particle amplitudes of non-interacting fermions in entangled states

TL;DR

This paper demonstrates that computing multi-particle scattering amplitudes for pairwise entangled fermions is #P-hard, linking it to the complexity of calculating matrix permanents, with implications for quantum computing demonstrations.

Contribution

It shows that pairwise entanglement of fermions suffices for #P-hardness in computing scattering amplitudes, extending complexity results to fermionic systems.

Findings

Computing multi-particle amplitudes for entangled fermions is #P-hard.

Exterior products of two-forms can express fermionic amplitudes.

The complexity of fermionic amplitudes relates to matrix permanents.

Abstract

Noninteracting 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 multi-particle 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 two. In particular, a permanent of a NxN matrix may be expressed as an exterior product of N^2 two-forms of rank two in dimension 2N^2, which establishes the #P-hardness of the latter.