 Finiteness of the projective Clifford group

For fixed $n$, a Clifford gate permutes the finite Pauli group by conjugation. Its action is determined by the images of $X_j,Z_j$, so there are finitely many possible such actions. If two unitary operators induce the same action, their quotient commutes with every $Z_j$ and is diagonal in the computational basis. Commutation with every $X_j$ equates diagonal entries at bit strings differing in one position; those flips connect all strings, so the quotient is a scalar. A unitary scalar is a global phase. Thus the Clifford group modulo global phase is finite and cannot approximate every unitary arbitrarily well. This proves the obstruction for any fixed number of wires, rather than relying solely on a one-qubit picture.^[ref]