Core Concepts
Quantum states prepared by Clifford circuits with fewer than n/2 non-Clifford single-qubit gates cannot be computationally pseudorandom.
Abstract
The key insights are:
The output state |ψ⟩ of a t-doped Clifford circuit (a Clifford circuit with at most t non-Clifford single-qubit gates) has stabilizer dimension at least n - 2t. This means the Weyl distribution qψ is supported on a subspace of dimension at most 2n - 2.
Haar-random states have minimal stabilizer dimension (0) with overwhelming probability. In contrast, the stabilizer dimension of the output of a t-doped Clifford circuit decreases by at most 2 per non-Clifford gate.
This allows distinguishing the output of a t-doped Clifford circuit from a Haar-random state by sampling from the Weyl distribution qψ and checking the dimension of the subspace spanned by the samples.
As a consequence, any family of Clifford circuits that produces an ensemble of computationally pseudorandom quantum states must use at least n/2 non-Clifford single-qubit gates. This bound is tight up to constant factors under plausible computational assumptions.