A gate for a Quantum Computer ∣ϕ⟩⊗∣ψ⟩ ⟼ ∣ψ⟩⊗∣ϕ⟩\ket{\phi} \otimes \ket{\psi} \;\longmapsto\; \ket{\psi} \otimes \ket{\phi}∣ϕ⟩⊗∣ψ⟩⟼∣ψ⟩⊗∣ϕ⟩ i.e., SWAP=CNOT1→2CNOT2→1CNOT1→2SWAP = CNOT_{1\rightarrow 2}CNOT_{2\rightarrow 1}CNOT_{1\rightarrow 2}SWAP=CNOT1→2CNOT2→1CNOT1→2 SWAP=(1000001001000001)SWAP=\begin{pmatrix} 1&0&0&0 \\ 0&0&1&0 \\ 0&1&0&0 \\ 0&0&0&1 \end{pmatrix}SWAP=1000001001000001Universality Theorem