Given a random Clifford state on $L$ qubits (defined as an infinite depth Clifford circuit acting on the zero state), what depth Clifford circuit is required to approximate this state to a given accuracy $\epsilon$.
I am essentially looking for a Kitaev Solovay theorem restricted to Clifford circuits. For a Haar random unitary on $L$ qubits, it is know that the required circuit depth to approximate it to a given accuracy scales as $2^L$. Is this also the case for random Clifford unitary gates on $L$ qubits?