The Clifford group is the set of quantum gates that map Pauli operators (I, X, Y, Z and their tensor products) to other Pauli operators under conjugation. The single-qubit Clifford gates include the Hadamard (H), Phase (S), and Pauli gates (X, Y, Z), along with all their compositions. The two-qubit CNOT gate is also a Clifford gate. Together, these gates can create highly entangled states and complex quantum circuits — yet they can be efficiently simulated on a classical computer.
This remarkable result, known as the Gottesman-Knill theorem, states that any circuit composed entirely of Clifford gates, starting from a computational basis state, and ending with measurements in the computational basis, can be efficiently simulated classically using the stabilizer formalism. This means that superposition and entanglement alone are not sufficient for quantum advantage — the non-Clifford T gate (or equivalent) is the ingredient that makes quantum computation genuinely hard to simulate.
In practice, Clifford gates are the "easy" gates in fault-tolerant quantum computing. They can be implemented transversally on stabilizer codes like the surface code, meaning they are applied independently to each physical qubit without spreading errors between code blocks. This makes Clifford operations cheap in terms of error correction overhead. The asymmetry between cheap Clifford gates and expensive non-Clifford gates (primarily the T gate, which requires magic state distillation) is a defining feature of fault-tolerant quantum computing architecture.