I'm interested in whether there is some known result about Clifford circuits being insufficent for classical computation.
Aaronson and Gottesman in their 2004 paper Improved Simulation of Stabilizer Circuits say the following...
Since stabilizer circuits are a generalization of CNOT circuits, it is obvious that Gottesman-Knill is ⊕L- hard (i.e. any ⊕L problem can be reduced to it). Our result says that Gottesman-Knill is in ⊕L. Intuitively, this means that any stabilizer circuit can be simulated efficiently using CNOT gates alone the additional availability of Hadamard and phase gates gives stabilizer circuits at most a polynomial advantage. In our view, this surprising fact helps to explain the Gottesman-Knill theorem, by providing strong evidence that stabilizer circuits are not even universal for classical computation
They prove this $\oplus L$ completeness result in section V of their paper.
Is it now proven that Clifford circuits are not sufficent for classical computation? In the paper Aaronson and Gottesman provide evidence for this but not a full proof as far as I know.
