A Clifford gate is a unitary operation that normalizes the Pauli group: conjugating any Pauli operator by it produces another Pauli operator. Hadamard, phase, and controlled- gates generate the Clifford operations.
A strong classical simulation computes a quantum circuit's specified output probabilities in classical polynomial time to the requested polynomial precision. This is stronger than weak simulation, which only samples from the output distribution.
To compute a single-qubit output probability of a Clifford circuit, propagate its measured Pauli observable backward through the circuit. Every conjugation remains a tensor-product Pauli, whose expectation on a product state factors into efficiently computable one-qubit expectations.
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In quantum mechanics and quantum information theory, the Pauli group is a set of important matrices related to the Pauli operators, which play a crucial role in the formulation of quantum gates and quantum error correction. The Pauli group on \( n \) qubits, denoted as \( \mathcal{P}_n \), consists of all \( n \)-qubit operators that can be expressed as the tensor products of the Pauli operators, up to a phase factor.