Qubit code

A convenient and often considered error set is the Pauli error or Pauli string basis. For a single qubit, this set consists of products of powers of the Pauli matrices \begin{align} X=\begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix}\,\,\text{ and }\,\,Z=\begin{pmatrix}1 & 0\\ 0 & -1 \end{pmatrix}~. \end{align} For multiple qubits, error set elements are tensor products of elements of the single-qubit error set.

The Pauli error set is a unitary and Hermitian basis for linear operators on the multi-qubit Hilbert space that is orthonormal under the Hilbert-Schmidt inner product; it is a prototypical nice error basis [1]. The distance associated with this set is often the minimum weight of a Pauli string that implements a nontrivial logical operation in the code. The minimum weight of a Pauli error that has a non-zero expectation value for some code basis state is called the diagonal distance [2]. Codes whose distance is greater than the diagonal distance are degenerate.