  It is well known that a quantum circuit on $N$ qubits composed of Clifford
gates with the addition of $k$ non Clifford gates can be simulated on a
classical computer by an algorithm scaling as $\text{poly}(N)\exp(k)$[1]. We
show that, for a quantum circuit to simulate quantum chaotic behavior, it is
both necessary and sufficient that $k=O(N)$. This result implies the
impossibility of simulating quantum chaos on a classical computer.
