= 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.<The Heisenberg Representation of Quantum Computers by Daniel Gottesman>[Section 3 and Table 1, Pauli normalizer and Clifford generators]{ref}
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