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Clifford gate set contains $\{CNOT, S, H\}\,.$

If the input state is a pure state, the Clifford circuit can be easily simulated by the Gottesman-Knill theorem. I wonder whether the statement still holds when the input is a mixed state.

Besides, what's the state-of-the-art paper about Stabilizer formalism?

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Your statement is not quite correct. A Clifford circuit with a stabilizer state as input can be efficiently simulated. With arbitrary pure states, we get universal quantum computing via magic state injection.

That being said, Clifford simulation can be straightforwardly extended to mixed states which are proportional to projectors onto stabilizer codes (here we simply propagate the generators of the code under the Clifford circuits).

These mixed states are uniform convex combinations of stabilizer states (take a basis of the code). More generally, we can simulate any mixed state which is a convex combination of stabilizer states, provided that we can sample efficiently from the underlying probability distribution. Indeed, in this case, we can just perform a Monte-Carlo simulation by sampling a stabilizer state and propagating it through the circuit.

In principle, there are also more mixed states that can be simulated efficiently. Stabilizer simulation can be extended to arbitrary states in multiple ways, leading to different sets of simulable states. A comprehensive overview of these methods is given by Seddon et al.. In particular, certain mixed states with low "magic" can still be simulated in polynomial time.

Note: In general, we have to distinguish between weak simulation (i.e. sampling from the outcome distribution of the circuit) or strong simulation (computing or estimating expectation values). Extension of stabilizer simulation to arbitrary states is usually limited to the estimation of expectation values.

PS: There's no good paper / review / book on stabilizer topics.

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