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I think your hierarchy collapses, or at least would never get beyond $P$, following the top-line results of Bravyi and Gosset. Bravyi and Gosset's paper gives an algorithm to classically simulate a quantum circuit on $n$ qubits comprising $O(\mathrm{poly\:}n)$ Clifford gates and a constant number of $T$ gates - that is, polynomial in $n$ (although ...


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