3 The binary symplectic representation
Parity is linear algebra over \(\mathbb {F}_2\), and that observation is the engine of the whole subject. Forgetting phases sends \(\mathcal{P}_n\) onto \(\mathbb {F}_2^{2n}\), products become sums, and — by theorem 17 — commutation becomes orthogonality under a symplectic form. An abelian subgroup becomes a self-orthogonal subspace, and independence of generators becomes a rank computation.
The linear-algebra shadow of a Pauli operator: an \(n\)-qubit Pauli operator is sent to a vector in \(\mathbb {F}_2^{2n}\), recording in its first \(n\) coordinates which qubits carry an \(X\) component and in its last \(n\) which carry a \(Z\) component (so \(Y\) contributes to both). Products of Paulis become sums of vectors, which is what turns questions about the group \(\mathcal{P}_n\) into linear algebra over \(\mathbb {F}_2\).
Definition 14 is injective: a Pauli operator is determined by its symplectic vector. Phases are of course not recorded.
The pair of bits at position \(i\) determines the \(i\)-th single-qubit operator, since the four Paulis \(I, X, Z, Y\) are sent to the four distinct pairs \((0,0)\), \((1,0)\), \((0,1)\), \((1,1)\).
For \(u = (u_X \mid u_Z)\) and \(v = (v_X \mid v_Z)\) in \(\mathbb {F}_2^{2n}\), the symplectic form \(\langle u, v \rangle = u_X \cdot v_Z + u_Z \cdot v_X \in \mathbb {F}_2\).
Two elements of \(\mathcal{P}_n\) commute if and only if their symplectic vectors are orthogonal under definition 16. This is the bridge that lets the whole theory be done with linear algebra over \(\mathbb {F}_2\): an abelian subgroup becomes a self-orthogonal subspace.
By theorem 11 commutation is the parity of the number of locally anticommuting qubits. Qubit \(i\) contributes \(1\) to \(\langle u, v \rangle \) exactly when the two single-qubit operators there are distinct and neither is \(I\), so the symplectic product computes precisely that parity.
The check matrix of a list of \(\mathcal{P}_n\) elements: the matrix over \(\mathbb {F}_2\) whose rows are the symplectic vectors (definition 14) of the listed generators. For a CSS code it is block diagonal, carrying the two classical parity-check matrices \(H_X\) and \(H_Z\) on the diagonal.
The rows of the check matrix (definition 18) are linearly independent over \(\mathbb {F}_2\). This is the computable criterion used to discharge the independence obligation of definition 37.