We unpack the Clifford group C_n, its Pauli-normalizing property, and how a small set of gates—Hadamard, Phase (S), and CNOT—generate all Clifford operations used in quantum error correction. Learn how Clifford circuits keep Pauli errors in check, enabling fault-tolerant syndrome measurements, and why Gottesman–Knill shows they can be efficiently simulated on a classical computer. Then explore why true quantum advantage requires stepping outside the Clifford group with non-Clifford gates like the T gate (often via magic states), highlighting the essential boundary between what’s classically tractable and what remains uniquely quantum.

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