Universality of single quantum gates

σz = 1 0 0 −1 basis. We treat the 2-qubit Swap gate ↑↑ ↑↓ ↓↑ ↓↓ ↑↑ 1 0 0 0 ↑↓ 0 0 1 0 ↓↑ 0 1 0 0 ↓↓ 0 0 0 1 as inherent to the circuit model, i.e. the timelines of qubits can be permuted. Thus, for example, a given 2-qubit gate can always be applied to any pair of qubits and in either order. The proof of “universality” of a given set of gates, i.e. universality for the class BQP (polynomial time quantum computation), consists of two steps: (1) showing that such a gate set is dense in PU(2) for all n (= ] of qubits in system), and (2) checking polynomial efficiency, which is an exercise in the Kitaev-Solovay (K-T) algorithm [8]. It is known [2] that if the single-qubit gates alone are dense in the projective unitary group PU(2) = U(2)/U(1) ∼= SO(3), then adding any additional 2-qubit gate which is entangling makes the gate set universal (G “entangling” means there exists a vector φ⊗ ψ so that G(φ⊗ ψ) 6= φ′ ⊗ ψ′ for any φ′ and ψ′). Our theorem will also comprise these two aspects listed above but we will not comment on the efficiency aspect since this is by now routine and parallel to the discussion of Refs. [8, 9].