Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 324 / 2 / iv / Solution

At each qubit, two matrices from $\{I,X,Y,Z\}$ either commute or anticommute. Moving every $Q_i$ past the corresponding $P_i$ therefore giveswhere $r$ is the number of positions containing distinct nonidentity Pauli matrices. Thus two Pauli strings always commute or anticommute. If they anticommute and a vector $|\psi ⟩$ were stabilized by both, then $PQ|\psi ⟩=|\psi ⟩$ and $QP|\psi ⟩=|\psi ⟩$, contradicting $PQ=-QP$. Their common stabilizer subspace is consequently the zero subspace $\{0\}$.