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I wonder what is a Qiskit stabilizer and why we need it. Also I want to know the difference between stabilizer and extended stabilizer.

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These are related to simulation of a quantum device on a classical computer. If all the gates in a circuit are H, S, and CNOT, then the circuit can be simulated on a classical computer in polynomial time. This is what the stabilizer simulator does. If the circuit is composed mostly of H, S, and CNOT, but there is also a small number t of T gates, then the circuit can be simulated in time that polynomial in 2^t. This is what the extended stabilizer simulator does.

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