Pauli errors and error-basis expansion

Use the computational basis and the Pauli matrices $I,X,Y,Z$, with $Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}$. They are orthogonal under the trace inner product: multiplying two equal Paulis gives trace two, while distinct ones give trace zero. Therefore even a nonunitary two-by-two matrix has a unique expansion in these four operators. Complex coefficients are allowed. Hermitian matrices have real coefficients, but a Kraus operator describing one environmental outcome generally need not be Hermitian.

For several qubits, tensor products of the four matrices form an operator basis. The weight of a Pauli string counts its nonidentity factors, rather than the number of negative signs or imaginary coefficients. An interaction confined to one specified qubit expands in identity and the three Paulis on that site. A joint interaction on many sites can require high-weight strings. Finite basis size alone supplies no useful bound on those components or on their correlations across successive operations.

A coherent phase error is a unitary rotation with a definite relative phase between its identity and phase-flip components. Replacing it by a random phase flip discards interference terms in the density operator. The two descriptions can agree for a particular measured probability and disagree for a different observable or for repeated applications. A fixed small overrotation accumulates its angle on repetition; independently sampled faults instead accumulate probabilities according to their specified stochastic process.

A general channel on the retained qubit can be written as a sum over Kraus operators, each expanded in the Pauli basis. Trace preservation constrains the entire Kraus set. It does not turn the expansion coefficients of an individual operator into four probabilities. A Pauli channel is the special case with nonnegative weights multiplying Pauli conjugations. Such a model may be justified by deliberate randomization or an approximation, but the justification must accompany any numerical error-rate calculation.

If one recovery corrects every operator in an error set and all their linear combinations on a code, an unknown superposition of those errors is covered. This is why a code correcting arbitrary single-site Pauli errors can correct an arbitrary qubit error on one site. The requirement concerns one common recovery, including cross terms between errors; separately finding an inverse for each known error is insufficient when the actual error is unknown.

The conclusion stops at the modeled Hilbert space and error support. Leakage into an additional energy level is not an operator acting solely within a two-dimensional qubit space. Reset errors, correlated crosstalk, and environment memory require corresponding models. Nor does correcting one error imply correcting two simultaneous errors. Before counting failures, state whether locations are independent, whether the channel is coherent or stochastic, and whether initialization and readout are included. These assumptions become part of the eventual protection claim.