Lecture 4. Symplectic structure of Pauli group and Clifford group

In this Lecture we have continued to discuss the properties of the stabilizer formalism and proved a number of important structural properties. Any Pauli observable on $n$ qubits can be naturally encoded as a bit string of length $2n+1$. In such an encoding, the multiplication of Pauli operators corresponds to the summation of their corresponding vectors, and the change in phase is related to the symplectic structure on $\mathbb{Z}_2^{2n}$. Any stabilizer group $\mathcal{S} = \langle P_1,\dots , P_l\rangle$ can be represented as a stabilizer tableau of size $l\times (2n+1)$, with the commutation of the rows corresponding to the nulling of the symplectic form. Clifford group is the group of all unitary operations translating Pauli observables into Pauli observables; when acting on a stabilizer tableau, they correspond column operations. Any stabilizer group can be reduced to the form $\langle Z_1,\dots, Z_r\rangle$ by the action of Clifford unitaries. Each Clifford unitary can be encoded as an extended stabilizer tableau of size $2n\times (2n+1)$, which describes the action of Pauli groups, and is a symplectic matrix up to signs.