Analysis of the PFC ensemble on the non-distinct subspace

Develop an analysis of the PFC ensemble on the non-distinct (collision) subspace of (C^d)^{⊗ t} sufficient to determine whether the PFC ensemble forms a relative-error t-design with linear scaling in t, by characterizing its action beyond the distinct-string subspace.

Background

Their proofs rely on restricting attention to the subspace spanned by distinct computational basis tuples and showing strong design properties there. Extending to relative-error designs requires controlling the ensemble’s behavior when collisions occur among t inputs.

The authors explicitly state that their current methods fail to analyze the PFC ensemble on the non-distinct subspace, which is the key technical barrier to establishing linear-in-t relative-error design properties.

References

It may well be that the $PFC$ ensemble is a relative-error $t$-design with linear scaling in $t$, but our current analysis does not show this because we do not know how to analyse the $PFC$ ensemble on the non-distinct subspace.

— Simple constructions of linear-depth t-designs and pseudorandom unitaries (2404.12647 - Metger et al., 19 Apr 2024) in Section 7 (Discussion and future directions)