= Pauli expansion of a quantum error
{c}
{title2=$E=\sum_{a,b\in\mathbb F_2^n}c_{a,b}X^aZ^b$}
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 $X^aZ^b$, with scalar phases absorbed into the coefficients, its <computational basis> action is $E|x\rangle=\sum_{a,b}c_{a,b}(-1)^{b\cdot x}|x+a\rangle$, 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.
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