In Quantum invariants of 3-manifolds and quantum computation (“BK”), Bravyi and Kitaev constructed a universal set of gates {g1,g2,g3} for the Ising TQFT, the principle component of the Moore Read model for ν=5/2-FQHE, in an abstract context in which there were no restrictions on the global topology of the space-time. Gate g1 may be referred to as a π/4 phase gate. Gate g2 may be referred to as a controlled π phase gate. Gate g3, which has no real name, may be used for braiding.
For a laboratory device, the relevant space-time should embed in R2×R1. It seems almost certain that simply adding this constraint to the Bravyi/Kitaev context prevents the construction of a complete gate set. However, if a certain assumption is added to their model—i.e., that the topological changes 1, σ and ε can be distinguished on a simple (framed) loop γ in space-time—then {g1,g2,g3} may be realized in 2+1 dimensions. Projecting to the charge sectors 1, σ and ε extends the discussion of Topologically-Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State, by Das Sarma, Freedman, and Nayak (“DFN”), in which interferometry was proposed to distinguish the 1 and ε changes. By an extension of DFN, an interferometry measurement should be able to resolve the identity into the sum of three projectors: 1d={circumflex over (1)}⊕{circumflex over (σ)}⊕{circumflex over (ε)}. A further generalization, however, is needed.