Fibonacci Anyons and Quantum Computation
In Sect. 8.2, we mentioned the hope that, by using anyons to encode qubits, one could use braiding to transform anyon states in various ways, thereby enabling quantum computation. Two anyons are not sufficient for this purpose, because the braid group on two strands is abelian, whereas quantum computation needs non-commuting unitary transformations. In the case of Fibonacci anyons, the computation in the preceding subsection shows that the braiding transformation aT,T is diagonal in a suitable basis, so it splits into one-dimensional representations; this again shows its inadequacy for quantum computation.
With three Fibonacci anyons, the situation improves dramatically. In a suitable basis, the transformation that braids the first two of the three anyons, ат,т ® 1T, is still diagonal. The same goes for the transformation that braids the second and third anyons, but the suitable bases in these two cases are not the same. They differ by an associativity isomorphism a. More precisely, one is the conjugate of the other by ат,т,т. They do not commute.
In fact, such braiding transformations suffice to approximate arbitrary unitary transformations of the two-dimensional Hilbert space VT for т ®3. Furthermore, using six Fibonacci anyons to code two qubits, one can approximate, by braiding, the so- called “controlled not” gate, which, in combination with one-qubit gates, is sufficient to produce all unitary gates for an arbitrary number of qubits; that is, it is sufficient for quantum computation. We refer to [9, Sect.6] for these combinations of Fibonacci braidings.