PRU construction without the binary phase layer

Determine whether removing the binary phase operator from the constructions based on the distinct ensembles in the paper preserves pseudorandom-unitary security, in particular whether $P\bigotimes_{i=1}^{n} C_i$ is a pseudorandom unitary when the $C_i$ are independent uniformly random single-qubit Clifford gates.

Background

The paper proves that combining a random permutation and pseudorandom binary phase operator with a depth-one layer of independent single-qubit Clifford gates yields a pseudorandom unitary. A related construction, using a full random Clifford, removes the phase layer when a stronger non-plussed distinctness property is available.

It remains unresolved whether the depth-one single-qubit Clifford layer has this stronger property and consequently whether the phase layer can be omitted from the resulting construction.

References

In particular, is the depth-one ensemble of independent single-qubit Clifford gates non-plussed distinct, and would this imply that $P\bigotimes_{i=1}{n} C_i$ is a PRU?

— Distinctness threshold for pseudorandom unitaries  (2609.03065 - Raza et al., 2 Sep 2026) in Discussion and open questions, paragraph “Distinctness in stronger query models and beyond”