Efficient simulation with partial quantum Fourier transforms and quadratic phase gates

An abelian hypergroup is the algebraic structure underlying the hypergroup quantum systems considered in the paper. A partial quantum Fourier transform is a quantum Fourier transform applied only to selected components, and a quadratic phase gate is a gate implementing a quadratic phase function. A normalizer circuit is a circuit of the normalizer gates considered in the paper.

Partial-QFT and quadratic-phase simulation conjecture. There exist nontrivial families of abelian hypergroups for which the normalizer circuits of the paper's simulation theorem remain efficiently classically simulable when supplemented with partial quantum Fourier transforms and quadratic phase gates acting at arbitrary circuit locations.

The claim would extend efficient classical simulation from the paper's stated normalizer-circuit setting to circuits containing these additional operations at arbitrary positions. The source presents this as an unproved existence claim and gives no resolution evidence.