 Pauli expansion of a quantum error

The tensor products of single-qubit Pauli matrices form an orthogonal basis of the operator space on $n$ qubits, so every linear error operator has a Pauli expansion. Using $X^a{Z}^b$, with scalar phases absorbed into the coefficients, its computational basis action is $E|x⟩={\sum }_{a,b}{c}_{a,b}(-1{)}^{b\cdot x}|x+a⟩$, with binary arithmetic. This describes coherent errors as well as individual Pauli errors; it does not assume that the noise is a probabilistic Pauli channel. A physical noise map has Kraus operators of this form when the usual quantum channel description applies, for instance when the initial system and environment are uncorrelated.