Consequences for fermionic systems (determinant case)

Ascertain the implications for fermionic systems—such as fermionic linear optics and related classical simulation questions—of the fact that approximate multi‑linear formulas for the determinant require super‑polynomial size, given that the determinant naturally arises in fermionic statistics.

Background

The paper extends Raz’s super‑polynomial lower bounds from exact multi‑linear formulas to approximate ones, and notes that the same reasoning applies to the determinant. Because the determinant underlies amplitudes for fermionic statistics, understanding how these algebraic‑complexity barriers translate into resource and simulability statements for fermionic models is left unresolved.

References

Interestingly, both Raz's lower bound and \cref{thm:bordercompperinformal} are also valid for the determinant, i.e.\ (approximate) multi-linear formulas for the determinant must have super-polynomial size. While the permanent relates to bosonic statistics, the determinant appears naturally in fermionic statistics . We leave as an open question the consequences of our results for fermionic systems.

— Lower Bounds on Coherent State Rank  (2604.00766 - Cottier et al., 1 Apr 2026) in Discussion and open questions