Depth-1 distinctness for pseudorandom unitaries

The paper proves that a tensor-product ensemble of independently chosen single-qubit Clifford gates is $\operatorname{negl}(n)$-distinct when the number of queries is polynomial in $n$. This disproves the authors' conjecture that negligibly distinct ensembles must necessarily be entangling. As an application, the depth-$\log n$ global Clifford/unitary-$2$-design layer used in the $PFC$ pseudorandom-unitary construction can be replaced by a single depth-1 layer of local single-qubit $2$-designs while preserving the required distinctness property.

The paper proves that a tensor-product ensemble of independently chosen single-qubit Clifford gates is $\operatorname{negl}(n)$-distinct when the number of queries is polynomial in $n$. This disproves the authors' conjecture that negligibly distinct ensembles must necessarily be entangling. As an application, the depth-$\log n$ global Clifford/unitary-$2$-design layer used in the $PFC$ pseudorandom-unitary construction can be replaced by a single depth-1 layer of local single-qubit $2$-designs while preserving the required distinctness property.