Pseudorandomness from Subset States

We show it is possible to obtain quantum pseudorandomness and pseudoentanglement from random subset states -- i.e. quantum states which are equal superpositions over (pseudo)random subsets of strings. This answers an open question of Aaronson et al. [arXiv:2211.00747], who devised a similar construction augmented by pseudorandom phases. Our result follows from a direct calculation of the trace distance between $t$ copies of random subset states and the Haar measure, via the representation theory of the symmetric group. We show that the trace distance is negligibly small, as long as the subsets are of an appropriate size which is neither too big nor too small. In particular, we analyze the action of basis permutations on the symmetric subspace, and show that the largest component is described by the Johnson scheme: the double-cosets of the symmetric group $\mathbb{S}_N$ by the subgroup $\mathbb{S}_t \times \mathbb{S}_{N-t}$. The Gelfand pair property of this setting implies that the matrix eigenbasis coincides with the symmetric group irreducible blocks, with the largest eigenblock asymptotically approaching the Haar average. An immediate corollary of our result is that quantum pseudorandom and pseudoentangled state ensembles do not require relative phases.